Theorem fco 5328

Description: Composition of two mappings. (Contributed by NM, 29-Aug-1999.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)

Assertion

Ref	Expression
fco	|- ( ( F : B --> C /\ G : A --> B ) -> ( F o. G ) : A --> C )

Proof of Theorem fco

Step	Hyp	Ref	Expression
1		df-f 5167	. . 3 |- ( F : B --> C <-> ( F Fn B /\ ran F C_ C ) )
2		df-f 5167	. . 3 |- ( G : A --> B <-> ( G Fn A /\ ran G C_ B ) )
3		fnco 5271	. . . . . . 7 |- ( ( F Fn B /\ G Fn A /\ ran G C_ B ) -> ( F o. G ) Fn A )
4	3	3expib 1185	. . . . . 6 |- ( F Fn B -> ( ( G Fn A /\ ran G C_ B ) -> ( F o. G ) Fn A ) )
5	4	adantr 274	. . . . 5 |- ( ( F Fn B /\ ran F C_ C ) -> ( ( G Fn A /\ ran G C_ B ) -> ( F o. G ) Fn A ) )
6		rncoss 4849	. . . . . . 7 |- ran ( F o. G ) C_ ran F
7		sstr 3132	. . . . . . 7 |- ( ( ran ( F o. G ) C_ ran F /\ ran F C_ C ) -> ran ( F o. G ) C_ C )
8	6, 7	mpan 421	. . . . . 6 |- ( ran F C_ C -> ran ( F o. G ) C_ C )
9	8	adantl 275	. . . . 5 |- ( ( F Fn B /\ ran F C_ C ) -> ran ( F o. G ) C_ C )
10	5, 9	jctird 315	. . . 4 |- ( ( F Fn B /\ ran F C_ C ) -> ( ( G Fn A /\ ran G C_ B ) -> ( ( F o. G ) Fn A /\ ran ( F o. G ) C_ C ) ) )
11	10	imp 123	. . 3 |- ( ( ( F Fn B /\ ran F C_ C ) /\ ( G Fn A /\ ran G C_ B ) ) -> ( ( F o. G ) Fn A /\ ran ( F o. G ) C_ C ) )
12	1, 2, 11	syl2anb 289	. 2 |- ( ( F : B --> C /\ G : A --> B ) -> ( ( F o. G ) Fn A /\ ran ( F o. G ) C_ C ) )
13		df-f 5167	. 2 |- ( ( F o. G ) : A --> C <-> ( ( F o. G ) Fn A /\ ran ( F o. G ) C_ C ) )
14	12, 13	sylibr 133	1 |- ( ( F : B --> C /\ G : A --> B ) -> ( F o. G ) : A --> C )

Syntax hints:   -> wi 4   /\ wa 103   C_ wss 3098   ran crn 4580   o. ccom 4583   Fn wfn 5158   -->wf 5159

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-14 2128  ax-ext 2136  ax-sep 4078  ax-pow 4130  ax-pr 4164

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1740  df-eu 2006  df-mo 2007  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-ral 2437  df-rex 2438  df-v 2711  df-un 3102  df-in 3104  df-ss 3111  df-pw 3541  df-sn 3562  df-pr 3563  df-op 3565  df-br 3962  df-opab 4022  df-id 4248  df-xp 4585  df-rel 4586  df-cnv 4587  df-co 4588  df-dm 4589  df-rn 4590  df-fun 5165  df-fn 5166  df-f 5167

This theorem is referenced by:  fco2  5329  f1co  5380  foco  5395  mapen  6780  ctm  7039  enomnilem  7060  enmkvlem  7083  enwomnilem  7091  fnn0nninf  10314  fsumcl2lem  11272  fsumadd  11280  fprodmul  11465  algcvg  11896  cnco  12560  cnptopco  12561  lmtopcnp  12589  cnmpt11  12622  cnmpt21  12630  comet  12838  cnmet  12869  cncfco  12917  limccnpcntop  12983  dvcoapbr  13010  dvcjbr  13011  dvcj  13012