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Theorem fco2 6627
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 6624 . 2 (((𝐹𝐵):𝐵𝐶𝐺:𝐴𝐵) → ((𝐹𝐵) ∘ 𝐺):𝐴𝐶)
2 frn 6607 . . . . 5 (𝐺:𝐴𝐵 → ran 𝐺𝐵)
3 cores 6153 . . . . 5 (ran 𝐺𝐵 → ((𝐹𝐵) ∘ 𝐺) = (𝐹𝐺))
42, 3syl 17 . . . 4 (𝐺:𝐴𝐵 → ((𝐹𝐵) ∘ 𝐺) = (𝐹𝐺))
54adantl 482 . . 3 (((𝐹𝐵):𝐵𝐶𝐺:𝐴𝐵) → ((𝐹𝐵) ∘ 𝐺) = (𝐹𝐺))
65feq1d 6585 . 2 (((𝐹𝐵):𝐵𝐶𝐺:𝐴𝐵) → (((𝐹𝐵) ∘ 𝐺):𝐴𝐶 ↔ (𝐹𝐺):𝐴𝐶))
71, 6mpbid 231 1 (((𝐹𝐵):𝐵𝐶𝐺:𝐴𝐵) → (𝐹𝐺):𝐴𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1539  wss 3887  ran crn 5590  cres 5591  ccom 5593  wf 6429
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-fun 6435  df-fn 6436  df-f 6437
This theorem is referenced by:  fsuppcor  9163  prdsringd  19851  prdscrngd  19852  prds1  19853  prdstmdd  23275  prdsxmslem2  23685  eulerpartlemmf  32342  sseqf  32359  poimirlem9  35786  ftc1anclem3  35852  fco2d  41773
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