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Theorem f1cofveqaeqALT 7005
 Description: Alternate proof of f1cofveqaeq 7004, 1 essential step shorter, but having more bytes (305 versus 282). (Contributed by AV, 3-Feb-2021.) (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
f1cofveqaeqALT (((𝐹:𝐵1-1𝐶𝐺:𝐴1-1𝐵) ∧ (𝑋𝐴𝑌𝐴)) → ((𝐹‘(𝐺𝑋)) = (𝐹‘(𝐺𝑌)) → 𝑋 = 𝑌))

Proof of Theorem f1cofveqaeqALT
StepHypRef Expression
1 f1f 6557 . . . . 5 (𝐺:𝐴1-1𝐵𝐺:𝐴𝐵)
2 fvco3 6747 . . . . . . . 8 ((𝐺:𝐴𝐵𝑋𝐴) → ((𝐹𝐺)‘𝑋) = (𝐹‘(𝐺𝑋)))
32adantrr 716 . . . . . . 7 ((𝐺:𝐴𝐵 ∧ (𝑋𝐴𝑌𝐴)) → ((𝐹𝐺)‘𝑋) = (𝐹‘(𝐺𝑋)))
4 fvco3 6747 . . . . . . . 8 ((𝐺:𝐴𝐵𝑌𝐴) → ((𝐹𝐺)‘𝑌) = (𝐹‘(𝐺𝑌)))
54adantrl 715 . . . . . . 7 ((𝐺:𝐴𝐵 ∧ (𝑋𝐴𝑌𝐴)) → ((𝐹𝐺)‘𝑌) = (𝐹‘(𝐺𝑌)))
63, 5eqeq12d 2814 . . . . . 6 ((𝐺:𝐴𝐵 ∧ (𝑋𝐴𝑌𝐴)) → (((𝐹𝐺)‘𝑋) = ((𝐹𝐺)‘𝑌) ↔ (𝐹‘(𝐺𝑋)) = (𝐹‘(𝐺𝑌))))
76ex 416 . . . . 5 (𝐺:𝐴𝐵 → ((𝑋𝐴𝑌𝐴) → (((𝐹𝐺)‘𝑋) = ((𝐹𝐺)‘𝑌) ↔ (𝐹‘(𝐺𝑋)) = (𝐹‘(𝐺𝑌)))))
81, 7syl 17 . . . 4 (𝐺:𝐴1-1𝐵 → ((𝑋𝐴𝑌𝐴) → (((𝐹𝐺)‘𝑋) = ((𝐹𝐺)‘𝑌) ↔ (𝐹‘(𝐺𝑋)) = (𝐹‘(𝐺𝑌)))))
98adantl 485 . . 3 ((𝐹:𝐵1-1𝐶𝐺:𝐴1-1𝐵) → ((𝑋𝐴𝑌𝐴) → (((𝐹𝐺)‘𝑋) = ((𝐹𝐺)‘𝑌) ↔ (𝐹‘(𝐺𝑋)) = (𝐹‘(𝐺𝑌)))))
109imp 410 . 2 (((𝐹:𝐵1-1𝐶𝐺:𝐴1-1𝐵) ∧ (𝑋𝐴𝑌𝐴)) → (((𝐹𝐺)‘𝑋) = ((𝐹𝐺)‘𝑌) ↔ (𝐹‘(𝐺𝑋)) = (𝐹‘(𝐺𝑌))))
11 f1co 6568 . . 3 ((𝐹:𝐵1-1𝐶𝐺:𝐴1-1𝐵) → (𝐹𝐺):𝐴1-1𝐶)
12 f1veqaeq 7003 . . 3 (((𝐹𝐺):𝐴1-1𝐶 ∧ (𝑋𝐴𝑌𝐴)) → (((𝐹𝐺)‘𝑋) = ((𝐹𝐺)‘𝑌) → 𝑋 = 𝑌))
1311, 12sylan 583 . 2 (((𝐹:𝐵1-1𝐶𝐺:𝐴1-1𝐵) ∧ (𝑋𝐴𝑌𝐴)) → (((𝐹𝐺)‘𝑋) = ((𝐹𝐺)‘𝑌) → 𝑋 = 𝑌))
1410, 13sylbird 263 1 (((𝐹:𝐵1-1𝐶𝐺:𝐴1-1𝐵) ∧ (𝑋𝐴𝑌𝐴)) → ((𝐹‘(𝐺𝑋)) = (𝐹‘(𝐺𝑌)) → 𝑋 = 𝑌))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111   ∘ ccom 5527  ⟶wf 6328  –1-1→wf1 6329  ‘cfv 6332 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 5171  ax-nul 5178  ax-pr 5299 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-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3444  df-sbc 3723  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4805  df-br 5035  df-opab 5097  df-id 5429  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6291  df-fun 6334  df-fn 6335  df-f 6336  df-f1 6337  df-fv 6340 This theorem is referenced by: (None)
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