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Theorem f1cofveqaeqALT 6556
Description: Alternate proof of f1cofveqaeq 6555, 1 essential step shorter, but having more bytes (305 vs. 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 6139 . . . . 5 (𝐺:𝐴1-1𝐵𝐺:𝐴𝐵)
2 fvco3 6314 . . . . . . . 8 ((𝐺:𝐴𝐵𝑋𝐴) → ((𝐹𝐺)‘𝑋) = (𝐹‘(𝐺𝑋)))
32adantrr 753 . . . . . . 7 ((𝐺:𝐴𝐵 ∧ (𝑋𝐴𝑌𝐴)) → ((𝐹𝐺)‘𝑋) = (𝐹‘(𝐺𝑋)))
4 fvco3 6314 . . . . . . . 8 ((𝐺:𝐴𝐵𝑌𝐴) → ((𝐹𝐺)‘𝑌) = (𝐹‘(𝐺𝑌)))
54adantrl 752 . . . . . . 7 ((𝐺:𝐴𝐵 ∧ (𝑋𝐴𝑌𝐴)) → ((𝐹𝐺)‘𝑌) = (𝐹‘(𝐺𝑌)))
63, 5eqeq12d 2666 . . . . . 6 ((𝐺:𝐴𝐵 ∧ (𝑋𝐴𝑌𝐴)) → (((𝐹𝐺)‘𝑋) = ((𝐹𝐺)‘𝑌) ↔ (𝐹‘(𝐺𝑋)) = (𝐹‘(𝐺𝑌))))
76ex 449 . . . . 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 481 . . 3 ((𝐹:𝐵1-1𝐶𝐺:𝐴1-1𝐵) → ((𝑋𝐴𝑌𝐴) → (((𝐹𝐺)‘𝑋) = ((𝐹𝐺)‘𝑌) ↔ (𝐹‘(𝐺𝑋)) = (𝐹‘(𝐺𝑌)))))
109imp 444 . 2 (((𝐹:𝐵1-1𝐶𝐺:𝐴1-1𝐵) ∧ (𝑋𝐴𝑌𝐴)) → (((𝐹𝐺)‘𝑋) = ((𝐹𝐺)‘𝑌) ↔ (𝐹‘(𝐺𝑋)) = (𝐹‘(𝐺𝑌))))
11 f1co 6148 . . 3 ((𝐹:𝐵1-1𝐶𝐺:𝐴1-1𝐵) → (𝐹𝐺):𝐴1-1𝐶)
12 f1veqaeq 6554 . . 3 (((𝐹𝐺):𝐴1-1𝐶 ∧ (𝑋𝐴𝑌𝐴)) → (((𝐹𝐺)‘𝑋) = ((𝐹𝐺)‘𝑌) → 𝑋 = 𝑌))
1311, 12sylan 487 . 2 (((𝐹:𝐵1-1𝐶𝐺:𝐴1-1𝐵) ∧ (𝑋𝐴𝑌𝐴)) → (((𝐹𝐺)‘𝑋) = ((𝐹𝐺)‘𝑌) → 𝑋 = 𝑌))
1410, 13sylbird 250 1 (((𝐹:𝐵1-1𝐶𝐺:𝐴1-1𝐵) ∧ (𝑋𝐴𝑌𝐴)) → ((𝐹‘(𝐺𝑋)) = (𝐹‘(𝐺𝑌)) → 𝑋 = 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  ccom 5147  wf 5922  1-1wf1 5923  cfv 5926
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fv 5934
This theorem is referenced by: (None)
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