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Theorem domeng 8127
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 4791 . 2 (𝑦 = 𝐵 → (𝐴𝑦𝐴𝐵))
2 sseq2 3776 . . . 4 (𝑦 = 𝐵 → (𝑥𝑦𝑥𝐵))
32anbi2d 614 . . 3 (𝑦 = 𝐵 → ((𝐴𝑥𝑥𝑦) ↔ (𝐴𝑥𝑥𝐵)))
43exbidv 2002 . 2 (𝑦 = 𝐵 → (∃𝑥(𝐴𝑥𝑥𝑦) ↔ ∃𝑥(𝐴𝑥𝑥𝐵)))
5 vex 3354 . . 3 𝑦 ∈ V
65domen 8126 . 2 (𝐴𝑦 ↔ ∃𝑥(𝐴𝑥𝑥𝑦))
71, 4, 6vtoclbg 3418 1 (𝐵𝐶 → (𝐴𝐵 ↔ ∃𝑥(𝐴𝑥𝑥𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1631  wex 1852  wcel 2145  wss 3723   class class class wbr 4787  cen 8110  cdom 8111
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pr 5035  ax-un 7100
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-xp 5256  df-rel 5257  df-cnv 5258  df-dm 5260  df-rn 5261  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-en 8114  df-dom 8115
This theorem is referenced by:  undom  8208  mapdom1  8285  mapdom2  8291  domfi  8341  isfinite2  8378  unxpwdom  8654  domfin4  9339  pwfseq  9692  grudomon  9845  ufldom  21986  erdsze2lem1  31523
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