Theorem fmptcos 6066

Description: Composition of two functions expressed as mapping abstractions. (Contributed by NM, 22-May-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)

Hypotheses

Ref	Expression
fmptcof.1
fmptcof.2
fmptcof.3

Assertion

Ref	Expression
fmptcos

Distinct variable groups:   ,,   ,   ,S   ,

Proof of Theorem fmptcos

Dummy variable is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fmptcof.1	. 2
2		fmptcof.2	. 2
3		fmptcof.3	. . 3
4		nfcv 2619	. . . 4
5		nfcsb1v 3450	. . . 4
6		csbeq1a 3443	. . . 4
7	4, 5, 6	cbvmpt 4542	. . 3
8	3, 7	syl6eq 2514	. 2
9		csbeq1 3437	. 2
10	1, 2, 8, 9	fmptcof 6065	1

Syntax hints:  ->wi 4  =wceq 1395  e.wcel 1818  A.wral 2807  [_csb 3434  e.cmpt 4510  o.ccom 5008

This theorem is referenced by:  fmpt2co  6883  gsummptf1o  16990  divcncf  31686

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-8 1820  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3435  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-uni 4250  df-br 4453  df-opab 4511  df-mpt 4512  df-id 4800  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-fv 5601