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Theorem f1resveqaeq 34157
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 6910 . . . 4 (𝐶𝐴 → ((𝐹𝐴)‘𝐶) = (𝐹𝐶))
21ad2antrl 726 . . 3 (((𝐹𝐴):𝐴1-1𝐵 ∧ (𝐶𝐴𝐷𝐴)) → ((𝐹𝐴)‘𝐶) = (𝐹𝐶))
3 fvres 6910 . . . 4 (𝐷𝐴 → ((𝐹𝐴)‘𝐷) = (𝐹𝐷))
43ad2antll 727 . . 3 (((𝐹𝐴):𝐴1-1𝐵 ∧ (𝐶𝐴𝐷𝐴)) → ((𝐹𝐴)‘𝐷) = (𝐹𝐷))
52, 4eqeq12d 2748 . 2 (((𝐹𝐴):𝐴1-1𝐵 ∧ (𝐶𝐴𝐷𝐴)) → (((𝐹𝐴)‘𝐶) = ((𝐹𝐴)‘𝐷) ↔ (𝐹𝐶) = (𝐹𝐷)))
6 f1veqaeq 7258 . 2 (((𝐹𝐴):𝐴1-1𝐵 ∧ (𝐶𝐴𝐷𝐴)) → (((𝐹𝐴)‘𝐶) = ((𝐹𝐴)‘𝐷) → 𝐶 = 𝐷))
75, 6sylbird 259 1 (((𝐹𝐴):𝐴1-1𝐵 ∧ (𝐶𝐴𝐷𝐴)) → ((𝐹𝐶) = (𝐹𝐷) → 𝐶 = 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  cres 5678  1-1wf1 6540  cfv 6543
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-res 5688  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fv 6551
This theorem is referenced by:  f1resrcmplf1d  34159
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