Theorem fco 5491

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 5322	. . 3 |- ( F : B --> C <-> ( F Fn B /\ ran F C_ C ) )
2		df-f 5322	. . 3 |- ( G : A --> B <-> ( G Fn A /\ ran G C_ B ) )
3		fnco 5431	. . . . . . 7 |- ( ( F Fn B /\ G Fn A /\ ran G C_ B ) -> ( F o. G ) Fn A )
4	3	3expib 1230	. . . . . 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 4995	. . . . . . 7 |- ran ( F o. G ) C_ ran F
7		sstr 3232	. . . . . . 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 5322	. 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 3197   ran crn 4720   o. ccom 4723   Fn wfn 5313   -->wf 5314

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-br 4084  df-opab 4146  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-fun 5320  df-fn 5321  df-f 5322

This theorem is referenced by:  fco2  5492  f1co  5545  foco  5561  mapen  7015  ctm  7287  enomnilem  7316  enmkvlem  7339  enwomnilem  7347  fnn0nninf  10672  seqf1oglem2  10754  fsumcl2lem  11925  fsumadd  11933  fprodmul  12118  algcvg  12586  mhmco  13539  gsumwmhm  13547  gsumfzreidx  13890  gsumfzmhm  13896  psrnegcl  14663  cnco  14911  cnptopco  14912  lmtopcnp  14940  cnmpt11  14973  cnmpt21  14981  comet  15189  cnmet  15220  cnfldms  15226  cncfco  15281  limccnpcntop  15365  dvcoapbr  15397  dvcjbr  15398  dvcj  15399