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Theorem fcoss 45117
Description: Composition of two mappings. Similar to fco 6771, 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 6764 . 2 (𝜑𝐺:𝐷𝐴)
5 fco 6771 . 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 3976  ccom 5704  wf 6569
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-fun 6575  df-fn 6576  df-f 6577
This theorem is referenced by:  volicoff  45916  voliooicof  45917  ovolval2  46565  ovolval5lem2  46574  ovolval5lem3  46575  ovnovollem1  46577  ovnovollem2  46578
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