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Theorem fco2 6541
Description: Functionality of a composition with weakened out of domain condition on the first argument. (Contributed by Stefan O'Rear, 11-Mar-2015.)
Assertion
Ref Expression
fco2 (((𝐹𝐵):𝐵𝐶𝐺:𝐴𝐵) → (𝐹𝐺):𝐴𝐶)

Proof of Theorem fco2
StepHypRef Expression
1 fco 6538 . 2 (((𝐹𝐵):𝐵𝐶𝐺:𝐴𝐵) → ((𝐹𝐵) ∘ 𝐺):𝐴𝐶)
2 frn 6521 . . . . 5 (𝐺:𝐴𝐵 → ran 𝐺𝐵)
3 cores 6092 . . . . 5 (ran 𝐺𝐵 → ((𝐹𝐵) ∘ 𝐺) = (𝐹𝐺))
42, 3syl 17 . . . 4 (𝐺:𝐴𝐵 → ((𝐹𝐵) ∘ 𝐺) = (𝐹𝐺))
54adantl 485 . . 3 (((𝐹𝐵):𝐵𝐶𝐺:𝐴𝐵) → ((𝐹𝐵) ∘ 𝐺) = (𝐹𝐺))
65feq1d 6499 . 2 (((𝐹𝐵):𝐵𝐶𝐺:𝐴𝐵) → (((𝐹𝐵) ∘ 𝐺):𝐴𝐶 ↔ (𝐹𝐺):𝐴𝐶))
71, 6mpbid 235 1 (((𝐹𝐵):𝐵𝐶𝐺:𝐴𝐵) → (𝐹𝐺):𝐴𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1542  wss 3853  ran crn 5536  cres 5537  ccom 5539  wf 6345
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-sep 5177  ax-nul 5184  ax-pr 5306
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ral 3059  df-rex 3060  df-rab 3063  df-v 3402  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4222  df-if 4425  df-sn 4527  df-pr 4529  df-op 4533  df-br 5041  df-opab 5103  df-id 5439  df-xp 5541  df-rel 5542  df-cnv 5543  df-co 5544  df-dm 5545  df-rn 5546  df-res 5547  df-ima 5548  df-fun 6351  df-fn 6352  df-f 6353
This theorem is referenced by:  fsuppcor  8953  prdsringd  19496  prdscrngd  19497  prds1  19498  prdstmdd  22887  prdsxmslem2  23294  eulerpartlemmf  31924  sseqf  31941  poimirlem9  35441  ftc1anclem3  35507  fco2d  41359
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