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Theorem fcoss 39716
Description: Composition of two mappings. Similar to fco 6096, 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 6095 . 2 (𝜑𝐺:𝐷𝐴)
5 fco 6096 . 2 ((𝐹:𝐴𝐵𝐺:𝐷𝐴) → (𝐹𝐺):𝐷𝐵)
61, 4, 5syl2anc 694 1 (𝜑 → (𝐹𝐺):𝐷𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wss 3607  ccom 5147  wf 5922
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-fun 5928  df-fn 5929  df-f 5930
This theorem is referenced by:  volicoff  40530  voliooicof  40531  ovolval2  41179  ovolval5lem2  41188  ovnovollem1  41191  ovnovollem2  41192
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