Theorem fncofn 5835

Description: Composition of a function with domain and a function as a function with domain. Generalization of fnco 5442. (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 5429	. . . 4 |- ( F Fn A -> Fun F )
2		funco 5368	. . . 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 5359	. 2 |- ( ( F Fn A /\ Fun G ) -> ( F o. G ) Fn dom ( F o. G ) )
5		fndm 5431	. . . . . . 7 |- ( F Fn A -> dom F = A )
6	5	adantr 276	. . . . . 6 |- ( ( F Fn A /\ Fun G ) -> dom F = A )
7	6	eqcomd 2236	. . . . 5 |- ( ( F Fn A /\ Fun G ) -> A = dom F )
8	7	imaeq2d 5078	. . . 4 |- ( ( F Fn A /\ Fun G ) -> ( `' G " A ) = ( `' G " dom F ) )
9		dmco 5247	. . . 4 |- dom ( F o. G ) = ( `' G " dom F )
10	8, 9	eqtr4di 2281	. . 3 |- ( ( F Fn A /\ Fun G ) -> ( `' G " A ) = dom ( F o. G ) )
11	10	fneq2d 5423	. 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 1397   `'ccnv 4726   dom cdm 4727   "cima 4730   o. ccom 4731   Fun wfun 5322   Fn wfn 5323

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

This theorem is referenced by:  fcof  5836