Theorem fco 4573

Description: Composition of two mappings. (The proof was 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		fnco 4521	. . . . . 6 |- ((F Fn B /\ G Fn A /\ ran G C_ B) -> (F o. G) Fn A)
2	1	3expib 1070	. . . . 5 |- (F Fn B -> ((G Fn A /\ ran G C_ B) -> (F o. G) Fn A))
3	2	adantr 425	. . . 4 |- ((F Fn B /\ ran F C_ C) -> ((G Fn A /\ ran G C_ B) -> (F o. G) Fn A))
4		rncoss 4213	. . . . . 6 |- ran ( F o. G) C_ ran F
5		sstr 2625	. . . . . 6 |- ((ran ( F o. G) C_ ran F /\ ran F C_ C) -> ran ( F o. G) C_ C)
6	4, 5	mpan 759	. . . . 5 |- (ran F C_ C -> ran ( F o. G) C_ C)
7	6	adantl 424	. . . 4 |- ((F Fn B /\ ran F C_ C) -> ran ( F o. G) C_ C)
8	3, 7	jctird 663	. . 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)))
9	8	imp 377	. 2 |- (((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))
10		df-f 4010	. . 3 |- (F:B-->C <-> (F Fn B /\ ran F C_ C))
11		df-f 4010	. . 3 |- (G:A-->B <-> (G Fn A /\ ran G C_ B))
12	10, 11	anbi12i 540	. 2 |- ((F:B-->C /\ G:A-->B) <-> ((F Fn B /\ ran F C_ C) /\ (G Fn A /\ ran G C_ B)))
13		df-f 4010	. 2 |- ((F o. G):A-->C <-> ((F o. G) Fn A /\ ran ( F o. G) C_ C))
14	9, 12, 13	3imtr4i 236	1 |- ((F:B-->C /\ G:A-->B) -> (F o. G):A-->C)

Syntax hints:   -> wi 3   /\ wa 240   C_ wss 2593  ran crn 3987   o. ccom 3990   Fn wfn 3993  -->wf 3994

This theorem is referenced by:  f1co 4612  foco 4628  mapenlem1 5583  mapenlem2 5584  ac6lem 5916  uzrdgfnuzi 7718  ruclem17 8795  cnco 9045  cnpco 9046  cnmetba 9181  cnmet 9182  cncfmet 9183  remetba 9187  imsdf 9652  lnocoi 9757  sincolem 10014  hocofi 11329  homco1 11364  homco2 11538  hmopco 11585  opsqrlem1 11711  opsqrlem6 11716  pjinvari 11764  algrf 13739  algr0 13740  algrp1 13742  algcvg 13744  mapmapmap 14486  injsurinj 14487  ghomco 16040  rnghomco 16128

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-fun 4008  df-fn 4009  df-f 4010

Copyright terms: Public domain