Theorem fcof 5862

Description: Composition of a function with domain and codomain and a function as a function with domain and codomain. Generalization of fco 5526. (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 5355	. . 3 |- ( F : A --> B <-> ( F Fn A /\ ran F C_ B ) )
2		fncofn 5861	. . . . . . 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 5027	. . . . . . 7 |- ran ( F o. G ) C_ ran F
6		sstr 3245	. . . . . . 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 5355	. 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 3210   `'ccnv 4747   ran crn 4749   "cima 4751   o. ccom 4752   Fun wfun 5345   Fn wfn 5346   -->wf 5347

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4227  ax-pow 4286  ax-pr 4321

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2814  df-un 3214  df-in 3216  df-ss 3223  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-br 4109  df-opab 4171  df-id 4413  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-fun 5353  df-fn 5354  df-f 5355

This theorem is referenced by: (None)