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Theorem domeng 8642
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 5057 . 2 (𝑦 = 𝐵 → (𝐴𝑦𝐴𝐵))
2 sseq2 3927 . . . 4 (𝑦 = 𝐵 → (𝑥𝑦𝑥𝐵))
32anbi2d 632 . . 3 (𝑦 = 𝐵 → ((𝐴𝑥𝑥𝑦) ↔ (𝐴𝑥𝑥𝐵)))
43exbidv 1929 . 2 (𝑦 = 𝐵 → (∃𝑥(𝐴𝑥𝑥𝑦) ↔ ∃𝑥(𝐴𝑥𝑥𝐵)))
5 vex 3412 . . 3 𝑦 ∈ V
65domen 8641 . 2 (𝐴𝑦 ↔ ∃𝑥(𝐴𝑥𝑥𝑦))
71, 4, 6vtoclbg 3483 1 (𝐵𝐶 → (𝐴𝐵 ↔ ∃𝑥(𝐴𝑥𝑥𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1543  wex 1787  wcel 2110  wss 3866   class class class wbr 5053  cen 8623  cdom 8624
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-11 2158  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322  ax-un 7523
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-opab 5116  df-xp 5557  df-rel 5558  df-cnv 5559  df-dm 5561  df-rn 5562  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-en 8627  df-dom 8628
This theorem is referenced by:  undom  8733  mapdom1  8811  mapdom2  8817  domfi  8896  isfinite2  8929  unxpwdom  9205  djuinf  9802  domfin4  9925  pwfseq  10278  grudomon  10431  ufldom  22859  erdsze2lem1  32878
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