Theorem domeng 8752

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	⊢ (𝐵 ∈ 𝐶 → (𝐴 ≼ 𝐵 ↔ ∃𝑥(𝐴 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐵)))

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.

Step	Hyp	Ref	Expression
1		breq2 5078	. 2 ⊢ (𝑦 = 𝐵 → (𝐴 ≼ 𝑦 ↔ 𝐴 ≼ 𝐵))
2		sseq2 3947	. . . 4 ⊢ (𝑦 = 𝐵 → (𝑥 ⊆ 𝑦 ↔ 𝑥 ⊆ 𝐵))
3	2	anbi2d 629	. . 3 ⊢ (𝑦 = 𝐵 → ((𝐴 ≈ 𝑥 ∧ 𝑥 ⊆ 𝑦) ↔ (𝐴 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐵)))
4	3	exbidv 1924	. 2 ⊢ (𝑦 = 𝐵 → (∃𝑥(𝐴 ≈ 𝑥 ∧ 𝑥 ⊆ 𝑦) ↔ ∃𝑥(𝐴 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐵)))
5		vex 3436	. . 3 ⊢ 𝑦 ∈ V
6	5	domen 8751	. 2 ⊢ (𝐴 ≼ 𝑦 ↔ ∃𝑥(𝐴 ≈ 𝑥 ∧ 𝑥 ⊆ 𝑦))
7	1, 4, 6	vtoclbg 3507	1 ⊢ (𝐵 ∈ 𝐶 → (𝐴 ≼ 𝐵 ↔ ∃𝑥(𝐴 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐵)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539  ∃wex 1782   ∈ wcel 2106   ⊆ wss 3887   class class class wbr 5074   ≈ cen 8730   ≼ cdom 8731

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-11 2154  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-xp 5595  df-rel 5596  df-cnv 5597  df-dm 5599  df-rn 5600  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-en 8734  df-dom 8735

This theorem is referenced by:  undomOLD  8847  mapdom1  8929  mapdom2  8935  domfi  8975  isfinite2  9072  unxpwdom  9348  djuinf  9944  domfin4  10067  pwfseq  10420  grudomon  10573  ufldom  23113  erdsze2lem1  33165