Theorem fnco 5494

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 5481	. . . 4 |- ( F Fn A -> Fun F )
2		fnfun 5481	. . . 4 |- ( G Fn B -> Fun G )
3		funco 5420	. . . 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 1048	. 2 |- ( ( F Fn A /\ G Fn B /\ ran G C_ A ) -> Fun ( F o. G ) )
6		fndm 5483	. . . . . . 7 |- ( F Fn A -> dom F = A )
7	6	sseq2d 3278	. . . . . 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 5056	. . . . 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 1047	. . 3 |- ( ( F Fn A /\ G Fn B /\ ran G C_ A ) -> dom ( F o. G ) = dom G )
12		fndm 5483	. . . 4 |- ( G Fn B -> dom G = B )
13	12	3ad2ant2 1050	. . 3 |- ( ( F Fn A /\ G Fn B /\ ran G C_ A ) -> dom G = B )
14	11, 13	eqtrd 2271	. 2 |- ( ( F Fn A /\ G Fn B /\ ran G C_ A ) -> dom ( F o. G ) = B )
15		df-fn 5383	. 2 |- ( ( F o. G ) Fn B <-> ( Fun ( F o. G ) /\ dom ( F o. G ) = B ) )
16	5, 14, 15	sylanbrc 421	1 |- ( ( F Fn A /\ G Fn B /\ ran G C_ A ) -> ( F o. G ) Fn B )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   /\ w3a 1009   = wceq 1402   C_ wss 3220   dom cdm 4776   ran crn 4777   o. ccom 4780   Fun wfun 5374   Fn wfn 5375

This proof depends on 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 4251  ax-pow 4313  ax-pr 4348

This proof 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 3690  df-sn 3715  df-pr 3716  df-op 3718  df-br 4132  df-opab 4194  df-id 4440  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-fun 5382  df-fn 5383

This theorem is used by:  fco  5555  fnfco  5567  updjudhcoinlf  7426  updjudhcoinrg  7427  prdsinvlem  14303  upxp  15498  uptx  15500