Theorem fnco 5326

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 5315	. . . 4 |- ( F Fn A -> Fun F )
2		fnfun 5315	. . . 4 |- ( G Fn B -> Fun G )
3		funco 5258	. . . 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 1017	. 2 |- ( ( F Fn A /\ G Fn B /\ ran G C_ A ) -> Fun ( F o. G ) )
6		fndm 5317	. . . . . . 7 |- ( F Fn A -> dom F = A )
7	6	sseq2d 3187	. . . . . 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 4900	. . . . 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 1016	. . 3 |- ( ( F Fn A /\ G Fn B /\ ran G C_ A ) -> dom ( F o. G ) = dom G )
12		fndm 5317	. . . 4 |- ( G Fn B -> dom G = B )
13	12	3ad2ant2 1019	. . 3 |- ( ( F Fn A /\ G Fn B /\ ran G C_ A ) -> dom G = B )
14	11, 13	eqtrd 2210	. 2 |- ( ( F Fn A /\ G Fn B /\ ran G C_ A ) -> dom ( F o. G ) = B )
15		df-fn 5221	. 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 978   = wceq 1353   C_ wss 3131   dom cdm 4628   ran crn 4629   o. ccom 4632   Fun wfun 5212   Fn wfn 5213

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-br 4006  df-opab 4067  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-fun 5220  df-fn 5221

This theorem is referenced by:  fco  5383  fnfco  5392  updjudhcoinlf  7081  updjudhcoinrg  7082  upxp  13857  uptx  13859