Theorem dmin 2537

Description: The domain of an intersection belong to the intersection of domains. Theorem 6 of [Suppes] p. 60.

Assertion

Ref	Expression
dmin	⊢ dom (A ∩ B) ⊆ (dom A ∩ dom B)

Proof of Theorem dmin

Step	Hyp	Ref	Expression
1		19.40 773	. . 3 ⊢ (∃y(⟨x, y⟩ ∈ A ∧ ⟨x, y⟩ ∈ B) → (∃y⟨x, y⟩ ∈ A ∧ ∃y⟨x, y⟩ ∈ B))
2		visset 1350	. . . . 5 ⊢ x ∈ V
3	2	eldm2 2528	. . . 4 ⊢ (x ∈ dom (A ∩ B) ↔ ∃y⟨x, y⟩ ∈ (A ∩ B))
4		elin 1635	. . . . 5 ⊢ (⟨x, y⟩ ∈ (A ∩ B) ↔ (⟨x, y⟩ ∈ A ∧ ⟨x, y⟩ ∈ B))
5	4	biex 733	. . . 4 ⊢ (∃y⟨x, y⟩ ∈ (A ∩ B) ↔ ∃y(⟨x, y⟩ ∈ A ∧ ⟨x, y⟩ ∈ B))
6	3, 5	bitr 151	. . 3 ⊢ (x ∈ dom (A ∩ B) ↔ ∃y(⟨x, y⟩ ∈ A ∧ ⟨x, y⟩ ∈ B))
7		elin 1635	. . . 4 ⊢ (x ∈ (dom A ∩ dom B) ↔ (x ∈ dom A ∧ x ∈ dom B))
8	2	eldm2 2528	. . . . 5 ⊢ (x ∈ dom A ↔ ∃y⟨x, y⟩ ∈ A)
9	2	eldm2 2528	. . . . 5 ⊢ (x ∈ dom B ↔ ∃y⟨x, y⟩ ∈ B)
10	8, 9	anbi12i 369	. . . 4 ⊢ ((x ∈ dom A ∧ x ∈ dom B) ↔ (∃y⟨x, y⟩ ∈ A ∧ ∃y⟨x, y⟩ ∈ B))
11	7, 10	bitr 151	. . 3 ⊢ (x ∈ (dom A ∩ dom B) ↔ (∃y⟨x, y⟩ ∈ A ∧ ∃y⟨x, y⟩ ∈ B))
12	1, 6, 11	3imtr4 192	. 2 ⊢ (x ∈ dom (A ∩ B) → x ∈ (dom A ∩ dom B))
13	12	ssriv 1508	1 ⊢ dom (A ∩ B) ⊆ (dom A ∩ dom B)

Syntax hints:   ∧ wa 196  ∃wex 678   ∈ wcel 1092   ∩ cin 1486   ⊆ wss 1487  ⟨cop 1810  dom cdm 

This theorem is referenced by:  rnin 2645  mapdom2lem 3388

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-in 1491  df-ss 1492  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-dm 2428

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