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Theorem f1resveqaeq 32480
 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 6664 . . . 4 (𝐶𝐴 → ((𝐹𝐴)‘𝐶) = (𝐹𝐶))
21ad2antrl 727 . . 3 (((𝐹𝐴):𝐴1-1𝐵 ∧ (𝐶𝐴𝐷𝐴)) → ((𝐹𝐴)‘𝐶) = (𝐹𝐶))
3 fvres 6664 . . . 4 (𝐷𝐴 → ((𝐹𝐴)‘𝐷) = (𝐹𝐷))
43ad2antll 728 . . 3 (((𝐹𝐴):𝐴1-1𝐵 ∧ (𝐶𝐴𝐷𝐴)) → ((𝐹𝐴)‘𝐷) = (𝐹𝐷))
52, 4eqeq12d 2814 . 2 (((𝐹𝐴):𝐴1-1𝐵 ∧ (𝐶𝐴𝐷𝐴)) → (((𝐹𝐴)‘𝐶) = ((𝐹𝐴)‘𝐷) ↔ (𝐹𝐶) = (𝐹𝐷)))
6 f1veqaeq 6993 . 2 (((𝐹𝐴):𝐴1-1𝐵 ∧ (𝐶𝐴𝐷𝐴)) → (((𝐹𝐴)‘𝐶) = ((𝐹𝐴)‘𝐷) → 𝐶 = 𝐷))
75, 6sylbird 263 1 (((𝐹𝐴):𝐴1-1𝐵 ∧ (𝐶𝐴𝐷𝐴)) → ((𝐹𝐶) = (𝐹𝐷) → 𝐶 = 𝐷))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111   ↾ cres 5521  –1-1→wf1 6321  ‘cfv 6324 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-res 5531  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fv 6332 This theorem is referenced by:  f1resrcmplf1d  32482
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