Theorem qsid 7396

Description: A set is equal to its quotient set mod converse epsilon. (Note: converse epsilon is not an equivalence relation.) (Contributed by NM, 13-Aug-1995.) (Revised by Mario Carneiro, 9-Jul-2014.)

Assertion

Ref	Expression
qsid

Proof of Theorem qsid

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

Step	Hyp	Ref	Expression
1		vex 3112	. . . . . . 7
2	1	ecid 7395	. . . . . 6
3	2	eqeq2i 2475	. . . . 5
4		equcom 1794	. . . . 5
5	3, 4	bitri 249	. . . 4
6	5	rexbii 2959	. . 3
7		vex 3112	. . . 4
8	7	elqs 7383	. . 3
9		risset 2982	. . 3
10	6, 8, 9	3bitr4i 277	. 2
11	10	eqriv 2453	1

Syntax hints:  =wceq 1395  e.wcel 1818  E.wrex 2808  cep 4794  `'ccnv 5003  [cec 7328  /.cqs 7329

This theorem is referenced by:  dfcnqs  9540

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-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  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-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-br 4453  df-opab 4511  df-eprel 4796  df-xp 5010  df-cnv 5012  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-ec 7332  df-qs 7336