Theorem eldmg 2529

Description: Domain membership. Theorem 4 of [Suppes] p. 59.

Assertion

Ref	Expression
eldmg	⊢ (A ∈ C → (A ∈ dom B ↔ ∃y⟨A, y⟩ ∈ B))

Distinct variable group(s):   y,A   y,B

Proof of Theorem eldmg

Step	Hyp	Ref	Expression
1		eleq1 1149	. 2 ⊢ (x = A → (x ∈ dom B ↔ A ∈ dom B))
2		opeq1 1876	. . . 4 ⊢ (x = A → ⟨x, y⟩ = ⟨A, y⟩)
3	2	eleq1d 1155	. . 3 ⊢ (x = A → (⟨x, y⟩ ∈ B ↔ ⟨A, y⟩ ∈ B))
4	3	biexdv 936	. 2 ⊢ (x = A → (∃y⟨x, y⟩ ∈ B ↔ ∃y⟨A, y⟩ ∈ B))
5		visset 1350	. . 3 ⊢ x ∈ V
6	5	eldm2 2528	. 2 ⊢ (x ∈ dom B ↔ ∃y⟨x, y⟩ ∈ B)
7	1, 4, 6	vtoclbg 1384	1 ⊢ (A ∈ C → (A ∈ dom B ↔ ∃y⟨A, y⟩ ∈ B))

Syntax hints:   → wi 2   ↔ wb 127  ∃wex 678   = wceq 1091   ∈ wcel 1092  ⟨cop 1810  dom cdm 2410

This theorem is referenced by:  dmfco 2864

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-un 1490  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-dm 2428

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