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Theorem domeng 7122
 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 4216 . 2
2 sseq2 3370 . . . 4
32anbi2d 685 . . 3
43exbidv 1636 . 2
5 vex 2959 . . 3
65domen 7121 . 2
71, 4, 6vtoclbg 3012 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   wa 359  wex 1550   wceq 1652   wcel 1725   wss 3320   class class class wbr 4212   cen 7106   cdom 7107 This theorem is referenced by:  undom  7196  mapdom1  7272  mapdom2  7278  domfi  7330  isfinite2  7365  unxpwdom  7557  domfin4  8191  pwfseq  8539  grudomon  8692  ufldom  17994  erdsze2lem1  24889 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403  ax-un 4701 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-xp 4884  df-rel 4885  df-cnv 4886  df-dm 4888  df-rn 4889  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-en 7110  df-dom 7111
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