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Theorem domeng 8513
 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 5035 . 2 (𝑦 = 𝐵 → (𝐴𝑦𝐴𝐵))
2 sseq2 3941 . . . 4 (𝑦 = 𝐵 → (𝑥𝑦𝑥𝐵))
32anbi2d 631 . . 3 (𝑦 = 𝐵 → ((𝐴𝑥𝑥𝑦) ↔ (𝐴𝑥𝑥𝐵)))
43exbidv 1922 . 2 (𝑦 = 𝐵 → (∃𝑥(𝐴𝑥𝑥𝑦) ↔ ∃𝑥(𝐴𝑥𝑥𝐵)))
5 vex 3444 . . 3 𝑦 ∈ V
65domen 8512 . 2 (𝐴𝑦 ↔ ∃𝑥(𝐴𝑥𝑥𝑦))
71, 4, 6vtoclbg 3517 1 (𝐵𝐶 → (𝐴𝐵 ↔ ∃𝑥(𝐴𝑥𝑥𝐵)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538  ∃wex 1781   ∈ wcel 2111   ⊆ wss 3881   class class class wbr 5031   ≈ cen 8496   ≼ cdom 8497 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pr 5296  ax-un 7448 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5032  df-opab 5094  df-xp 5526  df-rel 5527  df-cnv 5528  df-dm 5530  df-rn 5531  df-fn 6330  df-f 6331  df-f1 6332  df-fo 6333  df-f1o 6334  df-en 8500  df-dom 8501 This theorem is referenced by:  undom  8595  mapdom1  8673  mapdom2  8679  domfi  8730  isfinite2  8767  unxpwdom  9044  djuinf  9606  domfin4  9729  pwfseq  10082  grudomon  10235  ufldom  22581  erdsze2lem1  32599
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