Theorem fco 5441

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 5275	. . 3 |- ( F : B --> C <-> ( F Fn B /\ ran F C_ C ) )
2		df-f 5275	. . 3 |- ( G : A --> B <-> ( G Fn A /\ ran G C_ B ) )
3		fnco 5384	. . . . . . 7 |- ( ( F Fn B /\ G Fn A /\ ran G C_ B ) -> ( F o. G ) Fn A )
4	3	3expib 1209	. . . . . 6 |- ( F Fn B -> ( ( G Fn A /\ ran G C_ B ) -> ( F o. G ) Fn A ) )
5	4	adantr 276	. . . . 5 |- ( ( F Fn B /\ ran F C_ C ) -> ( ( G Fn A /\ ran G C_ B ) -> ( F o. G ) Fn A ) )
6		rncoss 4949	. . . . . . 7 |- ran ( F o. G ) C_ ran F
7		sstr 3201	. . . . . . 7 |- ( ( ran ( F o. G ) C_ ran F /\ ran F C_ C ) -> ran ( F o. G ) C_ C )
8	6, 7	mpan 424	. . . . . 6 |- ( ran F C_ C -> ran ( F o. G ) C_ C )
9	8	adantl 277	. . . . 5 |- ( ( F Fn B /\ ran F C_ C ) -> ran ( F o. G ) C_ C )
10	5, 9	jctird 317	. . . 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 124	. . 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 291	. 2 |- ( ( F : B --> C /\ G : A --> B ) -> ( ( F o. G ) Fn A /\ ran ( F o. G ) C_ C ) )
13		df-f 5275	. 2 |- ( ( F o. G ) : A --> C <-> ( ( F o. G ) Fn A /\ ran ( F o. G ) C_ C ) )
14	12, 13	sylibr 134	1 |- ( ( F : B --> C /\ G : A --> B ) -> ( F o. G ) : A --> C )

Syntax hints:   -> wi 4   /\ wa 104   C_ wss 3166   ran crn 4676   o. ccom 4679   Fn wfn 5266   -->wf 5267

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  df-f 5275

This theorem is referenced by:  fco2  5442  f1co  5493  foco  5509  mapen  6943  ctm  7211  enomnilem  7240  enmkvlem  7263  enwomnilem  7271  fnn0nninf  10583  seqf1oglem2  10665  fsumcl2lem  11709  fsumadd  11717  fprodmul  11902  algcvg  12370  mhmco  13322  gsumwmhm  13330  gsumfzreidx  13673  gsumfzmhm  13679  psrnegcl  14445  cnco  14693  cnptopco  14694  lmtopcnp  14722  cnmpt11  14755  cnmpt21  14763  comet  14971  cnmet  15002  cnfldms  15008  cncfco  15063  limccnpcntop  15147  dvcoapbr  15179  dvcjbr  15180  dvcj  15181