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Theorem domeng 6923
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 4092 . 2 (𝑦 = 𝐵 → (𝐴𝑦𝐴𝐵))
2 sseq2 3251 . . . 4 (𝑦 = 𝐵 → (𝑥𝑦𝑥𝐵))
32anbi2d 464 . . 3 (𝑦 = 𝐵 → ((𝐴𝑥𝑥𝑦) ↔ (𝐴𝑥𝑥𝐵)))
43exbidv 1873 . 2 (𝑦 = 𝐵 → (∃𝑥(𝐴𝑥𝑥𝑦) ↔ ∃𝑥(𝐴𝑥𝑥𝐵)))
5 vex 2805 . . 3 𝑦 ∈ V
65domen 6922 . 2 (𝐴𝑦 ↔ ∃𝑥(𝐴𝑥𝑥𝑦))
71, 4, 6vtoclbg 2865 1 (𝐵𝐶 → (𝐴𝐵 ↔ ∃𝑥(𝐴𝑥𝑥𝐵)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1397  wex 1540  wcel 2202  wss 3200   class class class wbr 4088  cen 6907  cdom 6908
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530
This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-xp 4731  df-rel 4732  df-cnv 4733  df-dm 4735  df-rn 4736  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-en 6910  df-dom 6911
This theorem is referenced by:  mapdom1g  7033
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