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Theorem fco2 6220
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 6219 . 2 (((𝐹𝐵):𝐵𝐶𝐺:𝐴𝐵) → ((𝐹𝐵) ∘ 𝐺):𝐴𝐶)
2 frn 6214 . . . . 5 (𝐺:𝐴𝐵 → ran 𝐺𝐵)
3 cores 5799 . . . . 5 (ran 𝐺𝐵 → ((𝐹𝐵) ∘ 𝐺) = (𝐹𝐺))
42, 3syl 17 . . . 4 (𝐺:𝐴𝐵 → ((𝐹𝐵) ∘ 𝐺) = (𝐹𝐺))
54adantl 473 . . 3 (((𝐹𝐵):𝐵𝐶𝐺:𝐴𝐵) → ((𝐹𝐵) ∘ 𝐺) = (𝐹𝐺))
65feq1d 6191 . 2 (((𝐹𝐵):𝐵𝐶𝐺:𝐴𝐵) → (((𝐹𝐵) ∘ 𝐺):𝐴𝐶 ↔ (𝐹𝐺):𝐴𝐶))
71, 6mpbid 222 1 (((𝐹𝐵):𝐵𝐶𝐺:𝐴𝐵) → (𝐹𝐺):𝐴𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wss 3715  ran crn 5267  cres 5268  ccom 5270  wf 6045
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-br 4805  df-opab 4865  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-fun 6051  df-fn 6052  df-f 6053
This theorem is referenced by:  fsuppcor  8474  prdsringd  18812  prdscrngd  18813  prds1  18814  prdstmdd  22128  prdsxmslem2  22535  eulerpartlemmf  30746  sseqf  30763  poimirlem9  33731  ftc1anclem3  33800  fco2d  38963
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