Theorem dmin 5903

Description: The domain of an intersection is included in the intersection of the domains. Theorem 6 of [Suppes] p. 60. (Contributed by NM, 15-Sep-2004.)

Assertion

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

Proof of Theorem dmin

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		19.40 1916	. . 3 ⊢ (∃𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵) → (∃𝑦⟨𝑥, 𝑦⟩ ∈ 𝐴 ∧ ∃𝑦⟨𝑥, 𝑦⟩ ∈ 𝐵))
2		vex 3459	. . . . 5 ⊢ 𝑥 ∈ V
3	2	eldm2 5893	. . . 4 ⊢ (𝑥 ∈ dom (𝐴 ∩ 𝐵) ↔ ∃𝑦⟨𝑥, 𝑦⟩ ∈ (𝐴 ∩ 𝐵))
4		elin 3922	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 ∩ 𝐵) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵))
5	4	exbii 1878	. . . 4 ⊢ (∃𝑦⟨𝑥, 𝑦⟩ ∈ (𝐴 ∩ 𝐵) ↔ ∃𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵))
6	3, 5	bitri 278	. . 3 ⊢ (𝑥 ∈ dom (𝐴 ∩ 𝐵) ↔ ∃𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵))
7		elin 3922	. . . 4 ⊢ (𝑥 ∈ (dom 𝐴 ∩ dom 𝐵) ↔ (𝑥 ∈ dom 𝐴 ∧ 𝑥 ∈ dom 𝐵))
8	2	eldm2 5893	. . . . 5 ⊢ (𝑥 ∈ dom 𝐴 ↔ ∃𝑦⟨𝑥, 𝑦⟩ ∈ 𝐴)
9	2	eldm2 5893	. . . . 5 ⊢ (𝑥 ∈ dom 𝐵 ↔ ∃𝑦⟨𝑥, 𝑦⟩ ∈ 𝐵)
10	8, 9	anbi12i 639	. . . 4 ⊢ ((𝑥 ∈ dom 𝐴 ∧ 𝑥 ∈ dom 𝐵) ↔ (∃𝑦⟨𝑥, 𝑦⟩ ∈ 𝐴 ∧ ∃𝑦⟨𝑥, 𝑦⟩ ∈ 𝐵))
11	7, 10	bitri 278	. . 3 ⊢ (𝑥 ∈ (dom 𝐴 ∩ dom 𝐵) ↔ (∃𝑦⟨𝑥, 𝑦⟩ ∈ 𝐴 ∧ ∃𝑦⟨𝑥, 𝑦⟩ ∈ 𝐵))
12	1, 6, 11	3imtr4i 295	. 2 ⊢ (𝑥 ∈ dom (𝐴 ∩ 𝐵) → 𝑥 ∈ (dom 𝐴 ∩ dom 𝐵))
13	12	ssriv 3942	1 ⊢ dom (𝐴 ∩ 𝐵) ⊆ (dom 𝐴 ∩ dom 𝐵)

Syntax hints:   ∧ wa 400  ∃wex 1809   ∈ wcel 2143   ∩ cin 3905   ⊆ wss 3906  ⟨cop 4596  dom cdm 5663

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-br 5111  df-dm 5673

This theorem is referenced by:  rninOLD  6146  psssdm2  18638  hauseqcn  34269