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Theorem fcoss 44583
Description: Composition of two mappings. Similar to fco 6747, 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 6740 . 2 (𝜑𝐺:𝐷𝐴)
5 fco 6747 . 2 ((𝐹:𝐴𝐵𝐺:𝐷𝐴) → (𝐹𝐺):𝐷𝐵)
61, 4, 5syl2anc 583 1 (𝜑 → (𝐹𝐺):𝐷𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wss 3947  ccom 5682  wf 6544
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5149  df-opab 5211  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-fun 6550  df-fn 6551  df-f 6552
This theorem is referenced by:  volicoff  45383  voliooicof  45384  ovolval2  46032  ovolval5lem2  46041  ovolval5lem3  46042  ovnovollem1  46044  ovnovollem2  46045
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