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Theorem domeng 7549
 Description: Dominance in terms of equinumerosity, with the sethood requirement expressed as an antecedent. Example 1 of [Enderton] p. 146. (Contributed by NM, 24-Apr-2004.)
Assertion
Ref Expression
domeng
Distinct variable groups:   ,   ,
Allowed substitution hint:   ()

Proof of Theorem domeng
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 breq2 4460 . 2
2 sseq2 3521 . . . 4
32anbi2d 703 . . 3
43exbidv 1715 . 2
5 vex 3112 . . 3
65domen 7548 . 2
71, 4, 6vtoclbg 3168 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 184   wa 369   wceq 1395  wex 1613   wcel 1819   wss 3471   class class class wbr 4456   cen 7532   cdom 7533 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695  ax-un 6591 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  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-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-xp 5014  df-rel 5015  df-cnv 5016  df-dm 5018  df-rn 5019  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-en 7536  df-dom 7537 This theorem is referenced by:  undom  7624  mapdom1  7701  mapdom2  7707  domfi  7760  isfinite2  7796  unxpwdom  8033  domfin4  8708  pwfseq  9059  grudomon  9212  ufldom  20589  erdsze2lem1  28844
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