Theorem fnco 5384

Description: Composition of two functions. (Contributed by NM, 22-May-2006.)

Assertion

Ref	Expression
fnco	|- ( ( F Fn A /\ G Fn B /\ ran G C_ A ) -> ( F o. G ) Fn B )

Proof of Theorem fnco

Step	Hyp	Ref	Expression
1		fnfun 5371	. . . 4 |- ( F Fn A -> Fun F )
2		fnfun 5371	. . . 4 |- ( G Fn B -> Fun G )
3		funco 5311	. . . 4 |- ( ( Fun F /\ Fun G ) -> Fun ( F o. G ) )
4	1, 2, 3	syl2an 289	. . 3 |- ( ( F Fn A /\ G Fn B ) -> Fun ( F o. G ) )
5	4	3adant3 1020	. 2 |- ( ( F Fn A /\ G Fn B /\ ran G C_ A ) -> Fun ( F o. G ) )
6		fndm 5373	. . . . . . 7 |- ( F Fn A -> dom F = A )
7	6	sseq2d 3223	. . . . . 6 |- ( F Fn A -> ( ran G C_ dom F <-> ran G C_ A ) )
8	7	biimpar 297	. . . . 5 |- ( ( F Fn A /\ ran G C_ A ) -> ran G C_ dom F )
9		dmcosseq 4950	. . . . 5 |- ( ran G C_ dom F -> dom ( F o. G ) = dom G )
10	8, 9	syl 14	. . . 4 |- ( ( F Fn A /\ ran G C_ A ) -> dom ( F o. G ) = dom G )
11	10	3adant2 1019	. . 3 |- ( ( F Fn A /\ G Fn B /\ ran G C_ A ) -> dom ( F o. G ) = dom G )
12		fndm 5373	. . . 4 |- ( G Fn B -> dom G = B )
13	12	3ad2ant2 1022	. . 3 |- ( ( F Fn A /\ G Fn B /\ ran G C_ A ) -> dom G = B )
14	11, 13	eqtrd 2238	. 2 |- ( ( F Fn A /\ G Fn B /\ ran G C_ A ) -> dom ( F o. G ) = B )
15		df-fn 5274	. 2 |- ( ( F o. G ) Fn B <-> ( Fun ( F o. G ) /\ dom ( F o. G ) = B ) )
16	5, 14, 15	sylanbrc 417	1 |- ( ( F Fn A /\ G Fn B /\ ran G C_ A ) -> ( F o. G ) Fn B )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 981   = wceq 1373   C_ wss 3166   dom cdm 4675   ran crn 4676   o. ccom 4679   Fun wfun 5265   Fn wfn 5266

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-br 4045  df-opab 4106  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-fun 5273  df-fn 5274

This theorem is referenced by:  fco  5441  fnfco  5450  updjudhcoinlf  7182  updjudhcoinrg  7183  prdsinvlem  13440  upxp  14744  uptx  14746