Theorem fncofn 5827

Description: Composition of a function with domain and a function as a function with domain. Generalization of fnco 5437. (Contributed by AV, 17-Sep-2024.)

Assertion

Ref	Expression
fncofn	|- ( ( F Fn A /\ Fun G ) -> ( F o. G ) Fn ( `' G " A ) )

Proof of Theorem fncofn

Step	Hyp	Ref	Expression
1		fnfun 5424	. . . 4 |- ( F Fn A -> Fun F )
2		funco 5364	. . . 4 |- ( ( Fun F /\ Fun G ) -> Fun ( F o. G ) )
3	1, 2	sylan 283	. . 3 |- ( ( F Fn A /\ Fun G ) -> Fun ( F o. G ) )
4	3	funfnd 5355	. 2 |- ( ( F Fn A /\ Fun G ) -> ( F o. G ) Fn dom ( F o. G ) )
5		fndm 5426	. . . . . . 7 |- ( F Fn A -> dom F = A )
6	5	adantr 276	. . . . . 6 |- ( ( F Fn A /\ Fun G ) -> dom F = A )
7	6	eqcomd 2235	. . . . 5 |- ( ( F Fn A /\ Fun G ) -> A = dom F )
8	7	imaeq2d 5074	. . . 4 |- ( ( F Fn A /\ Fun G ) -> ( `' G " A ) = ( `' G " dom F ) )
9		dmco 5243	. . . 4 |- dom ( F o. G ) = ( `' G " dom F )
10	8, 9	eqtr4di 2280	. . 3 |- ( ( F Fn A /\ Fun G ) -> ( `' G " A ) = dom ( F o. G ) )
11	10	fneq2d 5418	. 2 |- ( ( F Fn A /\ Fun G ) -> ( ( F o. G ) Fn ( `' G " A ) <-> ( F o. G ) Fn dom ( F o. G ) ) )
12	4, 11	mpbird 167	1 |- ( ( F Fn A /\ Fun G ) -> ( F o. G ) Fn ( `' G " A ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   `'ccnv 4722   dom cdm 4723   "cima 4726   o. ccom 4727   Fun wfun 5318   Fn wfn 5319

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2802  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-br 4087  df-opab 4149  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-fun 5326  df-fn 5327

This theorem is referenced by:  fcof  5828