Theorem fcoss 39402

Description: Composition of two mappings. Similar to fco 6058, but with a weaker condition on the domain of F. (Contributed by Glauco Siliprandi, 3-Mar-2021.)

Hypotheses

Ref	Expression
fcoss.f	|- ( ph -> F : A --> B )
fcoss.c	|- ( ph -> C C_ A )
fcoss.g	|- ( ph -> G : D --> C )

Assertion

Ref	Expression
fcoss	|- ( ph -> ( F o. G ) : D --> B )

Proof of Theorem fcoss

Step	Hyp	Ref	Expression
1		fcoss.f	. 2 |- ( ph -> F : A --> B )
2		fcoss.g	. . 3 |- ( ph -> G : D --> C )
3		fcoss.c	. . 3 |- ( ph -> C C_ A )
4	2, 3	fssd 6057	. 2 |- ( ph -> G : D --> A )
5		fco 6058	. 2 |- ( ( F : A --> B /\ G : D --> A ) -> ( F o. G ) : D --> B )
6	1, 4, 5	syl2anc 693	1 |- ( ph -> ( F o. G ) : D --> B )

Syntax hints:   -> wi 4   C_ wss 3574   o. ccom 5118   -->wf 5884

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-fun 5890  df-fn 5891  df-f 5892

This theorem is referenced by:  volicoff  40212  voliooicof  40213  ovolval2  40858  ovolval5lem2  40867  ovnovollem1  40870  ovnovollem2  40871