Theorem dmin 5909

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 1889	. . 3 ⊢ (∃𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵) → (∃𝑦⟨𝑥, 𝑦⟩ ∈ 𝐴 ∧ ∃𝑦⟨𝑥, 𝑦⟩ ∈ 𝐵))
2		vex 3478	. . . . 5 ⊢ 𝑥 ∈ V
3	2	eldm2 5899	. . . 4 ⊢ (𝑥 ∈ dom (𝐴 ∩ 𝐵) ↔ ∃𝑦⟨𝑥, 𝑦⟩ ∈ (𝐴 ∩ 𝐵))
4		elin 3963	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 ∩ 𝐵) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵))
5	4	exbii 1850	. . . 4 ⊢ (∃𝑦⟨𝑥, 𝑦⟩ ∈ (𝐴 ∩ 𝐵) ↔ ∃𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵))
6	3, 5	bitri 274	. . 3 ⊢ (𝑥 ∈ dom (𝐴 ∩ 𝐵) ↔ ∃𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵))
7		elin 3963	. . . 4 ⊢ (𝑥 ∈ (dom 𝐴 ∩ dom 𝐵) ↔ (𝑥 ∈ dom 𝐴 ∧ 𝑥 ∈ dom 𝐵))
8	2	eldm2 5899	. . . . 5 ⊢ (𝑥 ∈ dom 𝐴 ↔ ∃𝑦⟨𝑥, 𝑦⟩ ∈ 𝐴)
9	2	eldm2 5899	. . . . 5 ⊢ (𝑥 ∈ dom 𝐵 ↔ ∃𝑦⟨𝑥, 𝑦⟩ ∈ 𝐵)
10	8, 9	anbi12i 627	. . . 4 ⊢ ((𝑥 ∈ dom 𝐴 ∧ 𝑥 ∈ dom 𝐵) ↔ (∃𝑦⟨𝑥, 𝑦⟩ ∈ 𝐴 ∧ ∃𝑦⟨𝑥, 𝑦⟩ ∈ 𝐵))
11	7, 10	bitri 274	. . 3 ⊢ (𝑥 ∈ (dom 𝐴 ∩ dom 𝐵) ↔ (∃𝑦⟨𝑥, 𝑦⟩ ∈ 𝐴 ∧ ∃𝑦⟨𝑥, 𝑦⟩ ∈ 𝐵))
12	1, 6, 11	3imtr4i 291	. 2 ⊢ (𝑥 ∈ dom (𝐴 ∩ 𝐵) → 𝑥 ∈ (dom 𝐴 ∩ dom 𝐵))
13	12	ssriv 3985	1 ⊢ dom (𝐴 ∩ 𝐵) ⊆ (dom 𝐴 ∩ dom 𝐵)

Syntax hints:   ∧ wa 396  ∃wex 1781   ∈ wcel 2106   ∩ cin 3946   ⊆ wss 3947  ⟨cop 4633  dom cdm 5675

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-br 5148  df-dm 5685

This theorem is referenced by:  rnin  6143  psssdm2  18530  hauseqcn  32866