Theorem fmptcof 6065

Description: Version of fmptco 6064 where needn't be distinct from . (Contributed by NM, 27-Dec-2014.)

Hypotheses

Ref	Expression
fmptcof.1
fmptcof.2
fmptcof.3
fmptcof.4

Assertion

Ref	Expression
fmptcof

Distinct variable groups:   ,,   ,   ,S   ,   ,

Proof of Theorem fmptcof

Dummy variables are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fmptcof.1	. . . . 5
2		nfcsb1v 3450	. . . . . . 7
3	2	nfel1 2635	. . . . . 6
4		csbeq1a 3443	. . . . . . 7
5	4	eleq1d 2526	. . . . . 6
6	3, 5	rspc 3204	. . . . 5
7	1, 6	mpan9 469	. . . 4
8		fmptcof.2	. . . . 5
9		nfcv 2619	. . . . . 6
10	9, 2, 4	cbvmpt 4542	. . . . 5
11	8, 10	syl6eq 2514	. . . 4
12		fmptcof.3	. . . . 5
13		nfcv 2619	. . . . . 6
14		nfcsb1v 3450	. . . . . 6
15		csbeq1a 3443	. . . . . 6
16	13, 14, 15	cbvmpt 4542	. . . . 5
17	12, 16	syl6eq 2514	. . . 4
18		csbeq1 3437	. . . 4
19	7, 11, 17, 18	fmptco 6064	. . 3
20		nfcv 2619	. . . 4
21		nfcv 2619	. . . . 5
22	2, 21	nfcsb 3452	. . . 4
23	4	csbeq1d 3441	. . . 4
24	20, 22, 23	cbvmpt 4542	. . 3
25	19, 24	syl6eqr 2516	. 2
26		eqid 2457	. . . 4
27		nfcvd 2620	. . . . . 6
28		fmptcof.4	. . . . . 6
29	27, 28	csbiegf 3458	. . . . 5
30	29	ralimi 2850	. . . 4
31		mpteq12 4531	. . . 4
32	26, 30, 31	sylancr 663	. . 3
33	1, 32	syl 16	. 2
34	25, 33	eqtrd 2498	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:  fmptcos  6066  yonedalem3b  15548  gsumcom2  17003  evl1sca  18370  cnmptk1  20182  cnmpt1k  20183  cnmptkk  20184  cncfmpt1f  21417  copco  21518  pcoass  21524  sincn  22839  coscn  22840  lgseisenlem3  23626  fcomptf  27503  eulerpartgbij  28311  cncfcompt2  31702

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