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Theorem f1resveqaeq 34088
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 6911 . . . 4 (𝐶𝐴 → ((𝐹𝐴)‘𝐶) = (𝐹𝐶))
21ad2antrl 727 . . 3 (((𝐹𝐴):𝐴1-1𝐵 ∧ (𝐶𝐴𝐷𝐴)) → ((𝐹𝐴)‘𝐶) = (𝐹𝐶))
3 fvres 6911 . . . 4 (𝐷𝐴 → ((𝐹𝐴)‘𝐷) = (𝐹𝐷))
43ad2antll 728 . . 3 (((𝐹𝐴):𝐴1-1𝐵 ∧ (𝐶𝐴𝐷𝐴)) → ((𝐹𝐴)‘𝐷) = (𝐹𝐷))
52, 4eqeq12d 2749 . 2 (((𝐹𝐴):𝐴1-1𝐵 ∧ (𝐶𝐴𝐷𝐴)) → (((𝐹𝐴)‘𝐶) = ((𝐹𝐴)‘𝐷) ↔ (𝐹𝐶) = (𝐹𝐷)))
6 f1veqaeq 7256 . 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 397   = wceq 1542  wcel 2107  cres 5679  1-1wf1 6541  cfv 6544
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-res 5689  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fv 6552
This theorem is referenced by:  f1resrcmplf1d  34090
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