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Theorem fco2 6526
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 6524 . 2 (((𝐹𝐵):𝐵𝐶𝐺:𝐴𝐵) → ((𝐹𝐵) ∘ 𝐺):𝐴𝐶)
2 frn 6513 . . . . 5 (𝐺:𝐴𝐵 → ran 𝐺𝐵)
3 cores 6095 . . . . 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 6492 . 2 (((𝐹𝐵):𝐵𝐶𝐺:𝐴𝐵) → (((𝐹𝐵) ∘ 𝐺):𝐴𝐶 ↔ (𝐹𝐺):𝐴𝐶))
71, 6mpbid 233 1 (((𝐹𝐵):𝐵𝐶𝐺:𝐴𝐵) → (𝐹𝐺):𝐴𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wss 3933  ran crn 5549  cres 5550  ccom 5552  wf 6344
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-br 5058  df-opab 5120  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-fun 6350  df-fn 6351  df-f 6352
This theorem is referenced by:  fsuppcor  8855  prdsringd  19291  prdscrngd  19292  prds1  19293  prdstmdd  22659  prdsxmslem2  23066  eulerpartlemmf  31532  sseqf  31549  poimirlem9  34782  ftc1anclem3  34850  fco2d  40391
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