card_lt_of_card_biUnion_lt — Mathlib

English

For a regular cardinal c and hs: #s < c, the cardinal of the biUnion over s of t a h a is less than c iff for all a in s, #(t a h a) < c.

Русский

Для регулярного кардинала c и hs: #s < c, кардинал объединения по биUnion < c эквивалентно для каждого a в s: #(t a ha) < c.

LaTeX

$$$IsRegular(c) \land \#s < c \Rightarrow #(\bigcup_{a \in s} t(a, a)\lhd) < c \;\Leftrightarrow\; \forall a \in s, \#(t(a, a)) < c$$$

Lean4

theorem card_lt_of_card_biUnion_lt {α β : Type u} {s : Set α} {t : ∀ a ∈ s, Set β} {c : Cardinal}
    (h : #(⋃ a ∈ s, t a ‹_›) < c) (a : α) (ha : a ∈ s) : #(t a ha) < c :=
  by
  rw [biUnion_eq_iUnion] at h
  have := card_lt_of_card_iUnion_lt h
  simp_all only [iUnion_coe_set, Subtype.forall]

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