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Theorem f1resveqaeq 35099
Description: If a function restricted to a class is one-to-one, then for any two elements of the class, the values of the function at those elements are equal only if the two elements are the same element. (Contributed by BTernaryTau, 27-Sep-2023.)
Assertion
Ref Expression
f1resveqaeq (((𝐹𝐴):𝐴1-1𝐵 ∧ (𝐶𝐴𝐷𝐴)) → ((𝐹𝐶) = (𝐹𝐷) → 𝐶 = 𝐷))

Proof of Theorem f1resveqaeq
StepHypRef Expression
1 fvres 6925 . . . 4 (𝐶𝐴 → ((𝐹𝐴)‘𝐶) = (𝐹𝐶))
21ad2antrl 728 . . 3 (((𝐹𝐴):𝐴1-1𝐵 ∧ (𝐶𝐴𝐷𝐴)) → ((𝐹𝐴)‘𝐶) = (𝐹𝐶))
3 fvres 6925 . . . 4 (𝐷𝐴 → ((𝐹𝐴)‘𝐷) = (𝐹𝐷))
43ad2antll 729 . . 3 (((𝐹𝐴):𝐴1-1𝐵 ∧ (𝐶𝐴𝐷𝐴)) → ((𝐹𝐴)‘𝐷) = (𝐹𝐷))
52, 4eqeq12d 2753 . 2 (((𝐹𝐴):𝐴1-1𝐵 ∧ (𝐶𝐴𝐷𝐴)) → (((𝐹𝐴)‘𝐶) = ((𝐹𝐴)‘𝐷) ↔ (𝐹𝐶) = (𝐹𝐷)))
6 f1veqaeq 7277 . 2 (((𝐹𝐴):𝐴1-1𝐵 ∧ (𝐶𝐴𝐷𝐴)) → (((𝐹𝐴)‘𝐶) = ((𝐹𝐴)‘𝐷) → 𝐶 = 𝐷))
75, 6sylbird 260 1 (((𝐹𝐴):𝐴1-1𝐵 ∧ (𝐶𝐴𝐷𝐴)) → ((𝐹𝐶) = (𝐹𝐷) → 𝐶 = 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  cres 5687  1-1wf1 6558  cfv 6561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-res 5697  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fv 6569
This theorem is referenced by:  f1resrcmplf1d  35101
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