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Theorem f1cofveqaeqALT 7279
Description: Alternate proof of f1cofveqaeq 7278, 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 6804 . . . . 5 (𝐺:𝐴1-1𝐵𝐺:𝐴𝐵)
2 fvco3 7008 . . . . . . . 8 ((𝐺:𝐴𝐵𝑋𝐴) → ((𝐹𝐺)‘𝑋) = (𝐹‘(𝐺𝑋)))
32adantrr 717 . . . . . . 7 ((𝐺:𝐴𝐵 ∧ (𝑋𝐴𝑌𝐴)) → ((𝐹𝐺)‘𝑋) = (𝐹‘(𝐺𝑋)))
4 fvco3 7008 . . . . . . . 8 ((𝐺:𝐴𝐵𝑌𝐴) → ((𝐹𝐺)‘𝑌) = (𝐹‘(𝐺𝑌)))
54adantrl 716 . . . . . . 7 ((𝐺:𝐴𝐵 ∧ (𝑋𝐴𝑌𝐴)) → ((𝐹𝐺)‘𝑌) = (𝐹‘(𝐺𝑌)))
63, 5eqeq12d 2753 . . . . . 6 ((𝐺:𝐴𝐵 ∧ (𝑋𝐴𝑌𝐴)) → (((𝐹𝐺)‘𝑋) = ((𝐹𝐺)‘𝑌) ↔ (𝐹‘(𝐺𝑋)) = (𝐹‘(𝐺𝑌))))
76ex 412 . . . . 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 406 . 2 (((𝐹:𝐵1-1𝐶𝐺:𝐴1-1𝐵) ∧ (𝑋𝐴𝑌𝐴)) → (((𝐹𝐺)‘𝑋) = ((𝐹𝐺)‘𝑌) ↔ (𝐹‘(𝐺𝑋)) = (𝐹‘(𝐺𝑌))))
11 f1co 6815 . . 3 ((𝐹:𝐵1-1𝐶𝐺:𝐴1-1𝐵) → (𝐹𝐺):𝐴1-1𝐶)
12 f1veqaeq 7277 . . 3 (((𝐹𝐺):𝐴1-1𝐶 ∧ (𝑋𝐴𝑌𝐴)) → (((𝐹𝐺)‘𝑋) = ((𝐹𝐺)‘𝑌) → 𝑋 = 𝑌))
1311, 12sylan 580 . 2 (((𝐹:𝐵1-1𝐶𝐺:𝐴1-1𝐵) ∧ (𝑋𝐴𝑌𝐴)) → (((𝐹𝐺)‘𝑋) = ((𝐹𝐺)‘𝑌) → 𝑋 = 𝑌))
1410, 13sylbird 260 1 (((𝐹:𝐵1-1𝐶𝐺:𝐴1-1𝐵) ∧ (𝑋𝐴𝑌𝐴)) → ((𝐹‘(𝐺𝑋)) = (𝐹‘(𝐺𝑌)) → 𝑋 = 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  ccom 5689  wf 6557  1-1wf1 6558  cfv 6561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fv 6569
This theorem is referenced by: (None)
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