Theorem fcof 5836

Description: Composition of a function with domain and codomain and a function as a function with domain and codomain. Generalization of fco 5502. (Contributed by AV, 18-Sep-2024.)

Assertion

Ref	Expression
fcof	|- ( ( F : A --> B /\ Fun G ) -> ( F o. G ) : ( `' G " A ) --> B )

Proof of Theorem fcof

Step	Hyp	Ref	Expression
1		df-f 5332	. . 3 |- ( F : A --> B <-> ( F Fn A /\ ran F C_ B ) )
2		fncofn 5835	. . . . . . 7 |- ( ( F Fn A /\ Fun G ) -> ( F o. G ) Fn ( `' G " A ) )
3	2	ex 115	. . . . . 6 |- ( F Fn A -> ( Fun G -> ( F o. G ) Fn ( `' G " A ) ) )
4	3	adantr 276	. . . . 5 |- ( ( F Fn A /\ ran F C_ B ) -> ( Fun G -> ( F o. G ) Fn ( `' G " A ) ) )
5		rncoss 5005	. . . . . . 7 |- ran ( F o. G ) C_ ran F
6		sstr 3234	. . . . . . 7 |- ( ( ran ( F o. G ) C_ ran F /\ ran F C_ B ) -> ran ( F o. G ) C_ B )
7	5, 6	mpan 424	. . . . . 6 |- ( ran F C_ B -> ran ( F o. G ) C_ B )
8	7	adantl 277	. . . . 5 |- ( ( F Fn A /\ ran F C_ B ) -> ran ( F o. G ) C_ B )
9	4, 8	jctird 317	. . . 4 |- ( ( F Fn A /\ ran F C_ B ) -> ( Fun G -> ( ( F o. G ) Fn ( `' G " A ) /\ ran ( F o. G ) C_ B ) ) )
10	9	imp 124	. . 3 |- ( ( ( F Fn A /\ ran F C_ B ) /\ Fun G ) -> ( ( F o. G ) Fn ( `' G " A ) /\ ran ( F o. G ) C_ B ) )
11	1, 10	sylanb 284	. 2 |- ( ( F : A --> B /\ Fun G ) -> ( ( F o. G ) Fn ( `' G " A ) /\ ran ( F o. G ) C_ B ) )
12		df-f 5332	. 2 |- ( ( F o. G ) : ( `' G " A ) --> B <-> ( ( F o. G ) Fn ( `' G " A ) /\ ran ( F o. G ) C_ B ) )
13	11, 12	sylibr 134	1 |- ( ( F : A --> B /\ Fun G ) -> ( F o. G ) : ( `' G " A ) --> B )

Syntax hints:   -> wi 4   /\ wa 104   C_ wss 3199   `'ccnv 4726   ran crn 4728   "cima 4730   o. ccom 4731   Fun wfun 5322   Fn wfn 5323   -->wf 5324

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2204  ax-ext 2212  ax-sep 4208  ax-pow 4266  ax-pr 4301

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1810  df-eu 2081  df-mo 2082  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ral 2514  df-rex 2515  df-v 2803  df-un 3203  df-in 3205  df-ss 3212  df-pw 3655  df-sn 3676  df-pr 3677  df-op 3679  df-br 4090  df-opab 4152  df-id 4392  df-xp 4733  df-rel 4734  df-cnv 4735  df-co 4736  df-dm 4737  df-rn 4738  df-res 4739  df-ima 4740  df-fun 5330  df-fn 5331  df-f 5332

This theorem is referenced by: (None)