Theorem fncofn 5884

Description: Composition of a function with domain and a function as a function with domain. Generalization of fnco 5486. (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 5473	. . . 4 |- ( F Fn A -> Fun F )
2		funco 5412	. . . 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 5403	. 2 |- ( ( F Fn A /\ Fun G ) -> ( F o. G ) Fn dom ( F o. G ) )
5		fndm 5475	. . . . . . 7 |- ( F Fn A -> dom F = A )
6	5	adantr 276	. . . . . 6 |- ( ( F Fn A /\ Fun G ) -> dom F = A )
7	6	eqcomd 2244	. . . . 5 |- ( ( F Fn A /\ Fun G ) -> A = dom F )
8	7	imaeq2d 5121	. . . 4 |- ( ( F Fn A /\ Fun G ) -> ( `' G " A ) = ( `' G " dom F ) )
9		dmco 5291	. . . 4 |- dom ( F o. G ) = ( `' G " dom F )
10	8, 9	eqtr4di 2289	. . 3 |- ( ( F Fn A /\ Fun G ) -> ( `' G " A ) = dom ( F o. G ) )
11	10	fneq2d 5467	. 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 1402   `'ccnv 4768   dom cdm 4769   "cima 4772   o. ccom 4773   Fun wfun 5366   Fn wfn 5367

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 4244  ax-pow 4306  ax-pr 4341

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 3687  df-sn 3711  df-pr 3712  df-op 3714  df-br 4126  df-opab 4188  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-fun 5374  df-fn 5375

This theorem is referenced by:  fcof  5885