Theorem ecdmn0 7373

Description: A representative of a nonempty equivalence class belongs to the domain of the equivalence relation. (Contributed by NM, 15-Feb-1996.) (Revised by Mario Carneiro, 9-Jul-2014.)

Assertion

Ref	Expression
ecdmn0

Proof of Theorem ecdmn0

Dummy variable is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3118	. 2
2		n0 3794	. . 3
3		ecexr 7335	. . . 4
4	3	exlimiv 1722	. . 3
5	2, 4	sylbi 195	. 2
6		vex 3112	. . . . 5
7		elecg 7369	. . . . 5
8	6, 7	mpan 670	. . . 4
9	8	exbidv 1714	. . 3
10	2	a1i 11	. . 3
11		eldmg 5203	. . 3
12	9, 10, 11	3bitr4rd 286	. 2
13	1, 5, 12	pm5.21nii 353	1

Syntax hints:  <->wb 184  E.wex 1612  e.wcel 1818  =/=wne 2652  cvv 3109  c0 3784   class class class wbr 4452  domcdm 5004  [cec 7328

This theorem is referenced by:  ereldm  7374  elqsn0  7399  ecelqsdm  7400  eceqoveq  7435  divsfval  14944  sylow1lem5  16622  vitalilem2  22018  vitalilem3  22019

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-xp 5010  df-cnv 5012  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-ec 7332