Theorem dfoprab4f 6858

Description: Operation class abstraction expressed without existential quantifiers. (Unnecessary distinct variable restrictions were removed by David Abernethy, 19-Jun-2012.) (Contributed by NM, 20-Dec-2008.) (Revised by Mario Carneiro, 31-Aug-2015.)

Hypotheses

Ref	Expression
dfoprab4f.x
dfoprab4f.y
dfoprab4f.1

Assertion

Ref	Expression
dfoprab4f

Distinct variable groups:   ,,,   ,,,   ,,,   ,

Proof of Theorem dfoprab4f

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

Step	Hyp	Ref	Expression
1		nfv 1707	. . . . 5
2		dfoprab4f.x	. . . . . 6
3		nfs1v 2181	. . . . . 6
4	2, 3	nfbi 1934	. . . . 5
5	1, 4	nfim 1920	. . . 4
6		opeq1 4217	. . . . . 6
7	6	eqeq2d 2471	. . . . 5
8		sbequ12 1992	. . . . . 6
9	8	bibi2d 318	. . . . 5
10	7, 9	imbi12d 320	. . . 4
11		nfv 1707	. . . . . 6
12		dfoprab4f.y	. . . . . . 7
13		nfs1v 2181	. . . . . . 7
14	12, 13	nfbi 1934	. . . . . 6
15	11, 14	nfim 1920	. . . . 5
16		opeq2 4218	. . . . . . 7
17	16	eqeq2d 2471	. . . . . 6
18		sbequ12 1992	. . . . . . 7
19	18	bibi2d 318	. . . . . 6
20	17, 19	imbi12d 320	. . . . 5
21		dfoprab4f.1	. . . . 5
22	15, 20, 21	chvar 2013	. . . 4
23	5, 10, 22	chvar 2013	. . 3
24	23	dfoprab4 6857	. 2
25		nfv 1707	. . 3
26		nfv 1707	. . 3
27		nfv 1707	. . . 4
28	27, 3	nfan 1928	. . 3
29		nfv 1707	. . . 4
30	13	nfsb 2184	. . . 4
31	29, 30	nfan 1928	. . 3
32		eleq1 2529	. . . . 5
33		eleq1 2529	. . . . 5
34	32, 33	bi2anan9 873	. . . 4
35	18, 8	sylan9bbr 700	. . . 4
36	34, 35	anbi12d 710	. . 3
37	25, 26, 28, 31, 36	cbvoprab12 6371	. 2
38	24, 37	eqtr4i 2489	1

Syntax hints:  ->wi 4  <->wb 184  /\wa 369  =wceq 1395  F/wnf 1616  [wsb 1739  e.wcel 1818  <.cop 4035  {copab 4509  X.cxp 5002  {coprab 6297

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  ax-un 6592

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-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-iota 5556  df-fun 5595  df-fv 5601  df-oprab 6300  df-1st 6800  df-2nd 6801