Theorem fcof 5888

Description: Composition of a function with domain and codomain and a function as a function with domain and codomain. Generalization of fco 5550. (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 5379	. . 3 |- ( F : A --> B <-> ( F Fn A /\ ran F C_ B ) )
2		fncofn 5887	. . . . . . 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 5051	. . . . . . 7 |- ran ( F o. G ) C_ ran F
6		sstr 3256	. . . . . . 7 |- ( ( ran ( F o. G ) C_ ran F /\ ran F C_ B ) -> ran ( F o. G ) C_ B )
7	5, 6	mpan 428	. . . . . 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 5379	. 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 3220   `'ccnv 4771   ran crn 4773   "cima 4775   o. ccom 4776   Fun wfun 5369   Fn wfn 5370   -->wf 5371

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-pow 4309  ax-pr 4344

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-br 4129  df-opab 4191  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-fun 5377  df-fn 5378  df-f 5379

This theorem is referenced by: (None)