Theorem fco 5423

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 5262	. . 3 |- ( F : B --> C <-> ( F Fn B /\ ran F C_ C ) )
2		df-f 5262	. . 3 |- ( G : A --> B <-> ( G Fn A /\ ran G C_ B ) )
3		fnco 5366	. . . . . . 7 |- ( ( F Fn B /\ G Fn A /\ ran G C_ B ) -> ( F o. G ) Fn A )
4	3	3expib 1208	. . . . . 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 4936	. . . . . . 7 |- ran ( F o. G ) C_ ran F
7		sstr 3191	. . . . . . 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 5262	. 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 3157   ran crn 4664   o. ccom 4667   Fn wfn 5253   -->wf 5254

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-br 4034  df-opab 4095  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-fun 5260  df-fn 5261  df-f 5262

This theorem is referenced by:  fco2  5424  f1co  5475  foco  5491  mapen  6907  ctm  7175  enomnilem  7204  enmkvlem  7227  enwomnilem  7235  fnn0nninf  10530  seqf1oglem2  10612  fsumcl2lem  11563  fsumadd  11571  fprodmul  11756  algcvg  12216  mhmco  13122  gsumwmhm  13130  gsumfzreidx  13467  gsumfzmhm  13473  cnco  14457  cnptopco  14458  lmtopcnp  14486  cnmpt11  14519  cnmpt21  14527  comet  14735  cnmet  14766  cnfldms  14772  cncfco  14827  limccnpcntop  14911  dvcoapbr  14943  dvcjbr  14944  dvcj  14945