graph_fst_eq_zero_snd — Mathlib

English

If two pairs (x, x') and (y, y') lie in the graph of f and x = y, then x' = y'.

Русский

Если пары (x, x') и (y, y') принадлежат графу f и x = y, то x' = y'.

LaTeX

$$$\text{If }(x,x'),(y,y')\in \mathrm{graph}(f)\text{ and } x=y,\; x'=y'.$$$

Lean4

/-- The property that `f 0 = 0` in terms of the graph. -/
theorem graph_fst_eq_zero_snd (f : E →ₗ.[R] F) {x : E} {x' : F} (h : (x, x') ∈ f.graph) (hx : x = 0) : x' = 0 :=
  f.mem_graph_snd_inj h f.graph.zero_mem hx

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