Theorem fncofn 5862

Description: Composition of a function with domain and a function as a function with domain. Generalization of fnco 5466. (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 5453	. . . 4 |- ( F Fn A -> Fun F )
2		funco 5392	. . . 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 5383	. 2 |- ( ( F Fn A /\ Fun G ) -> ( F o. G ) Fn dom ( F o. G ) )
5		fndm 5455	. . . . . . 7 |- ( F Fn A -> dom F = A )
6	5	adantr 276	. . . . . 6 |- ( ( F Fn A /\ Fun G ) -> dom F = A )
7	6	eqcomd 2238	. . . . 5 |- ( ( F Fn A /\ Fun G ) -> A = dom F )
8	7	imaeq2d 5101	. . . 4 |- ( ( F Fn A /\ Fun G ) -> ( `' G " A ) = ( `' G " dom F ) )
9		dmco 5271	. . . 4 |- dom ( F o. G ) = ( `' G " dom F )
10	8, 9	eqtr4di 2283	. . 3 |- ( ( F Fn A /\ Fun G ) -> ( `' G " A ) = dom ( F o. G ) )
11	10	fneq2d 5447	. 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 1398   `'ccnv 4748   dom cdm 4749   "cima 4752   o. ccom 4753   Fun wfun 5346   Fn wfn 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 4228  ax-pow 4287  ax-pr 4322

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 2815  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-br 4110  df-opab 4172  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-fun 5354  df-fn 5355

This theorem is referenced by:  fcof  5863