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Theorem domeng 8958
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 (𝐵𝐶 → (𝐴𝐵 ↔ ∃𝑥(𝐴𝑥𝑥𝐵)))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hint:   𝐶(𝑥)

Proof of Theorem domeng
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 breq2 5153 . 2 (𝑦 = 𝐵 → (𝐴𝑦𝐴𝐵))
2 sseq2 4009 . . . 4 (𝑦 = 𝐵 → (𝑥𝑦𝑥𝐵))
32anbi2d 630 . . 3 (𝑦 = 𝐵 → ((𝐴𝑥𝑥𝑦) ↔ (𝐴𝑥𝑥𝐵)))
43exbidv 1925 . 2 (𝑦 = 𝐵 → (∃𝑥(𝐴𝑥𝑥𝑦) ↔ ∃𝑥(𝐴𝑥𝑥𝐵)))
5 vex 3479 . . 3 𝑦 ∈ V
65domen 8957 . 2 (𝐴𝑦 ↔ ∃𝑥(𝐴𝑥𝑥𝑦))
71, 4, 6vtoclbg 3560 1 (𝐵𝐶 → (𝐴𝐵 ↔ ∃𝑥(𝐴𝑥𝑥𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wex 1782  wcel 2107  wss 3949   class class class wbr 5149  cen 8936  cdom 8937
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-11 2155  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-xp 5683  df-rel 5684  df-cnv 5685  df-dm 5687  df-rn 5688  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-en 8940  df-dom 8941
This theorem is referenced by:  undomOLD  9060  mapdom1  9142  mapdom2  9148  domfi  9192  isfinite2  9301  unxpwdom  9584  djuinf  10183  domfin4  10306  pwfseq  10659  grudomon  10812  ufldom  23466  erdsze2lem1  34194
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