Theorem fnresfnco 41206

Description: Composition of two functions, similar to fnco 5999. (Contributed by Alexander van der Vekens, 25-Jul-2017.)

Assertion

Ref	Expression
fnresfnco	|- ( ( ( F |` ran G ) Fn ran G /\ G Fn B ) -> ( F o. G ) Fn B )

Proof of Theorem fnresfnco

Step	Hyp	Ref	Expression
1		fnfun 5988	. . 3 |- ( ( F |` ran G ) Fn ran G -> Fun ( F |` ran G ) )
2		fnfun 5988	. . 3 |- ( G Fn B -> Fun G )
3		funresfunco 41205	. . 3 |- ( ( Fun ( F |` ran G ) /\ Fun G ) -> Fun ( F o. G ) )
4	1, 2, 3	syl2an 494	. 2 |- ( ( ( F |` ran G ) Fn ran G /\ G Fn B ) -> Fun ( F o. G ) )
5		fndm 5990	. . . . . 6 |- ( ( F |` ran G ) Fn ran G -> dom ( F |` ran G ) = ran G )
6		dmres 5419	. . . . . . . 8 |- dom ( F |` ran G ) = ( ran G i^i dom F )
7	6	eqeq1i 2627	. . . . . . 7 |- ( dom ( F |` ran G ) = ran G <-> ( ran G i^i dom F ) = ran G )
8		df-ss 3588	. . . . . . 7 |- ( ran G C_ dom F <-> ( ran G i^i dom F ) = ran G )
9	7, 8	sylbb2 228	. . . . . 6 |- ( dom ( F |` ran G ) = ran G -> ran G C_ dom F )
10	5, 9	syl 17	. . . . 5 |- ( ( F |` ran G ) Fn ran G -> ran G C_ dom F )
11	10	adantr 481	. . . 4 |- ( ( ( F |` ran G ) Fn ran G /\ G Fn B ) -> ran G C_ dom F )
12		dmcosseq 5387	. . . 4 |- ( ran G C_ dom F -> dom ( F o. G ) = dom G )
13	11, 12	syl 17	. . 3 |- ( ( ( F |` ran G ) Fn ran G /\ G Fn B ) -> dom ( F o. G ) = dom G )
14		fndm 5990	. . . 4 |- ( G Fn B -> dom G = B )
15	14	adantl 482	. . 3 |- ( ( ( F |` ran G ) Fn ran G /\ G Fn B ) -> dom G = B )
16	13, 15	eqtrd 2656	. 2 |- ( ( ( F |` ran G ) Fn ran G /\ G Fn B ) -> dom ( F o. G ) = B )
17		df-fn 5891	. 2 |- ( ( F o. G ) Fn B <-> ( Fun ( F o. G ) /\ dom ( F o. G ) = B ) )
18	4, 16, 17	sylanbrc 698	1 |- ( ( ( F |` ran G ) Fn ran G /\ G Fn B ) -> ( F o. G ) Fn B )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   i^i cin 3573   C_ wss 3574   dom cdm 5114   ran crn 5115   |` cres 5116   o. ccom 5118   Fun wfun 5882   Fn wfn 5883

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-fun 5890  df-fn 5891

This theorem is referenced by:  funcoressn  41207