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Theorem fcoss 38397
 Description: Composition of two mappings. Similar to fco 5971, but with a weaker condition on the domain of 𝐹. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
Hypotheses
Ref Expression
fcoss.f (𝜑𝐹:𝐴𝐵)
fcoss.c (𝜑𝐶𝐴)
fcoss.g (𝜑𝐺:𝐷𝐶)
Assertion
Ref Expression
fcoss (𝜑 → (𝐹𝐺):𝐷𝐵)

Proof of Theorem fcoss
StepHypRef Expression
1 fcoss.f . 2 (𝜑𝐹:𝐴𝐵)
2 fcoss.g . . 3 (𝜑𝐺:𝐷𝐶)
3 fcoss.c . . 3 (𝜑𝐶𝐴)
42, 3fssd 5970 . 2 (𝜑𝐺:𝐷𝐴)
5 fco 5971 . 2 ((𝐹:𝐴𝐵𝐺:𝐷𝐴) → (𝐹𝐺):𝐷𝐵)
61, 4, 5syl2anc 691 1 (𝜑 → (𝐹𝐺):𝐷𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ⊆ wss 3540   ∘ ccom 5042  ⟶wf 5800 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-fun 5806  df-fn 5807  df-f 5808 This theorem is referenced by:  volicoff  38888  voliooicof  38889  ovolval2  39534  ovolval5lem2  39543  ovnovollem1  39546  ovnovollem2  39547
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