Theorem fnco 5383

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 5370	. . . 4 |- ( F Fn A -> Fun F )
2		fnfun 5370	. . . 4 |- ( G Fn B -> Fun G )
3		funco 5310	. . . 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 1019	. 2 |- ( ( F Fn A /\ G Fn B /\ ran G C_ A ) -> Fun ( F o. G ) )
6		fndm 5372	. . . . . . 7 |- ( F Fn A -> dom F = A )
7	6	sseq2d 3222	. . . . . 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 4949	. . . . 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 1018	. . 3 |- ( ( F Fn A /\ G Fn B /\ ran G C_ A ) -> dom ( F o. G ) = dom G )
12		fndm 5372	. . . 4 |- ( G Fn B -> dom G = B )
13	12	3ad2ant2 1021	. . 3 |- ( ( F Fn A /\ G Fn B /\ ran G C_ A ) -> dom G = B )
14	11, 13	eqtrd 2237	. 2 |- ( ( F Fn A /\ G Fn B /\ ran G C_ A ) -> dom ( F o. G ) = B )
15		df-fn 5273	. 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 980   = wceq 1372   C_ wss 3165   dom cdm 4674   ran crn 4675   o. ccom 4678   Fun wfun 5264   Fn wfn 5265

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 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rex 2489  df-v 2773  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-br 4044  df-opab 4105  df-id 4339  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-fun 5272  df-fn 5273

This theorem is referenced by:  fco  5440  fnfco  5449  updjudhcoinlf  7181  updjudhcoinrg  7182  prdsinvlem  13411  upxp  14715  uptx  14717