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Theorem f1cofveqaeqALT 7207
Description: Alternate proof of f1cofveqaeq 7206, 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 6739 . . . . 5 (𝐺:𝐴1-1𝐵𝐺:𝐴𝐵)
2 fvco3 6941 . . . . . . . 8 ((𝐺:𝐴𝐵𝑋𝐴) → ((𝐹𝐺)‘𝑋) = (𝐹‘(𝐺𝑋)))
32adantrr 716 . . . . . . 7 ((𝐺:𝐴𝐵 ∧ (𝑋𝐴𝑌𝐴)) → ((𝐹𝐺)‘𝑋) = (𝐹‘(𝐺𝑋)))
4 fvco3 6941 . . . . . . . 8 ((𝐺:𝐴𝐵𝑌𝐴) → ((𝐹𝐺)‘𝑌) = (𝐹‘(𝐺𝑌)))
54adantrl 715 . . . . . . 7 ((𝐺:𝐴𝐵 ∧ (𝑋𝐴𝑌𝐴)) → ((𝐹𝐺)‘𝑌) = (𝐹‘(𝐺𝑌)))
63, 5eqeq12d 2749 . . . . . 6 ((𝐺:𝐴𝐵 ∧ (𝑋𝐴𝑌𝐴)) → (((𝐹𝐺)‘𝑋) = ((𝐹𝐺)‘𝑌) ↔ (𝐹‘(𝐺𝑋)) = (𝐹‘(𝐺𝑌))))
76ex 414 . . . . 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 483 . . 3 ((𝐹:𝐵1-1𝐶𝐺:𝐴1-1𝐵) → ((𝑋𝐴𝑌𝐴) → (((𝐹𝐺)‘𝑋) = ((𝐹𝐺)‘𝑌) ↔ (𝐹‘(𝐺𝑋)) = (𝐹‘(𝐺𝑌)))))
109imp 408 . 2 (((𝐹:𝐵1-1𝐶𝐺:𝐴1-1𝐵) ∧ (𝑋𝐴𝑌𝐴)) → (((𝐹𝐺)‘𝑋) = ((𝐹𝐺)‘𝑌) ↔ (𝐹‘(𝐺𝑋)) = (𝐹‘(𝐺𝑌))))
11 f1co 6751 . . 3 ((𝐹:𝐵1-1𝐶𝐺:𝐴1-1𝐵) → (𝐹𝐺):𝐴1-1𝐶)
12 f1veqaeq 7205 . . 3 (((𝐹𝐺):𝐴1-1𝐶 ∧ (𝑋𝐴𝑌𝐴)) → (((𝐹𝐺)‘𝑋) = ((𝐹𝐺)‘𝑌) → 𝑋 = 𝑌))
1311, 12sylan 581 . 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 205  wa 397   = wceq 1542  wcel 2107  ccom 5638  wf 6493  1-1wf1 6494  cfv 6497
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fv 6505
This theorem is referenced by: (None)
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