Theorem domeng 6871

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 4066	. 2 ⊢ (𝑦 = 𝐵 → (𝐴 ≼ 𝑦 ↔ 𝐴 ≼ 𝐵))
2		sseq2 3228	. . . 4 ⊢ (𝑦 = 𝐵 → (𝑥 ⊆ 𝑦 ↔ 𝑥 ⊆ 𝐵))
3	2	anbi2d 464	. . 3 ⊢ (𝑦 = 𝐵 → ((𝐴 ≈ 𝑥 ∧ 𝑥 ⊆ 𝑦) ↔ (𝐴 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐵)))
4	3	exbidv 1851	. 2 ⊢ (𝑦 = 𝐵 → (∃𝑥(𝐴 ≈ 𝑥 ∧ 𝑥 ⊆ 𝑦) ↔ ∃𝑥(𝐴 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐵)))
5		vex 2782	. . 3 ⊢ 𝑦 ∈ V
6	5	domen 6870	. 2 ⊢ (𝐴 ≼ 𝑦 ↔ ∃𝑥(𝐴 ≈ 𝑥 ∧ 𝑥 ⊆ 𝑦))
7	1, 4, 6	vtoclbg 2842	1 ⊢ (𝐵 ∈ 𝐶 → (𝐴 ≼ 𝐵 ↔ ∃𝑥(𝐴 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐵)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1375  ∃wex 1518   ∈ wcel 2180   ⊆ wss 3177   class class class wbr 4062   ≈ cen 6855   ≼ cdom 6856

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 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-13 2182  ax-14 2183  ax-ext 2191  ax-sep 4181  ax-pow 4237  ax-pr 4272  ax-un 4501

This theorem depends on definitions:  df-bi 117  df-3an 985  df-tru 1378  df-nf 1487  df-sb 1789  df-eu 2060  df-mo 2061  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-ral 2493  df-rex 2494  df-v 2781  df-un 3181  df-in 3183  df-ss 3190  df-pw 3631  df-sn 3652  df-pr 3653  df-op 3655  df-uni 3868  df-br 4063  df-opab 4125  df-xp 4702  df-rel 4703  df-cnv 4704  df-dm 4706  df-rn 4707  df-fn 5297  df-f 5298  df-f1 5299  df-fo 5300  df-f1o 5301  df-en 6858  df-dom 6859

This theorem is referenced by:  mapdom1g  6976