MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  f1cofveqaeqALT Structured version   Visualization version   GIF version

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 6805 . . . . 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 2751 . . . . . 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 6816 . . 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 1537  wcel 2106  ccom 5693  wf 6559  1-1wf1 6560  cfv 6563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fv 6571
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator