Theorem fco 5383

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 5222	. . 3 |- ( F : B --> C <-> ( F Fn B /\ ran F C_ C ) )
2		df-f 5222	. . 3 |- ( G : A --> B <-> ( G Fn A /\ ran G C_ B ) )
3		fnco 5326	. . . . . . 7 |- ( ( F Fn B /\ G Fn A /\ ran G C_ B ) -> ( F o. G ) Fn A )
4	3	3expib 1206	. . . . . 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 4899	. . . . . . 7 |- ran ( F o. G ) C_ ran F
7		sstr 3165	. . . . . . 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 5222	. 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 3131   ran crn 4629   o. ccom 4632   Fn wfn 5213   -->wf 5214

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  df-f 5222

This theorem is referenced by:  fco2  5384  f1co  5435  foco  5450  mapen  6848  ctm  7110  enomnilem  7138  enmkvlem  7161  enwomnilem  7169  fnn0nninf  10439  fsumcl2lem  11408  fsumadd  11416  fprodmul  11601  algcvg  12050  mhmco  12879  cnco  13760  cnptopco  13761  lmtopcnp  13789  cnmpt11  13822  cnmpt21  13830  comet  14038  cnmet  14069  cncfco  14117  limccnpcntop  14183  dvcoapbr  14210  dvcjbr  14211  dvcj  14212