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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 (𝐴𝐵) ⊆ (dom 𝐴 ∩ dom 𝐵)

Proof of Theorem dmin
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 19.40 1882 . . 3 (∃𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵) → (∃𝑦𝑥, 𝑦⟩ ∈ 𝐴 ∧ ∃𝑦𝑥, 𝑦⟩ ∈ 𝐵))
2 vex 3474 . . . . 5 𝑥 ∈ V
32eldm2 5899 . . . 4 (𝑥 ∈ dom (𝐴𝐵) ↔ ∃𝑦𝑥, 𝑦⟩ ∈ (𝐴𝐵))
4 elin 3961 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ (𝐴𝐵) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵))
54exbii 1843 . . . 4 (∃𝑦𝑥, 𝑦⟩ ∈ (𝐴𝐵) ↔ ∃𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵))
63, 5bitri 275 . . 3 (𝑥 ∈ dom (𝐴𝐵) ↔ ∃𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵))
7 elin 3961 . . . 4 (𝑥 ∈ (dom 𝐴 ∩ dom 𝐵) ↔ (𝑥 ∈ dom 𝐴𝑥 ∈ dom 𝐵))
82eldm2 5899 . . . . 5 (𝑥 ∈ dom 𝐴 ↔ ∃𝑦𝑥, 𝑦⟩ ∈ 𝐴)
92eldm2 5899 . . . . 5 (𝑥 ∈ dom 𝐵 ↔ ∃𝑦𝑥, 𝑦⟩ ∈ 𝐵)
108, 9anbi12i 627 . . . 4 ((𝑥 ∈ dom 𝐴𝑥 ∈ dom 𝐵) ↔ (∃𝑦𝑥, 𝑦⟩ ∈ 𝐴 ∧ ∃𝑦𝑥, 𝑦⟩ ∈ 𝐵))
117, 10bitri 275 . . 3 (𝑥 ∈ (dom 𝐴 ∩ dom 𝐵) ↔ (∃𝑦𝑥, 𝑦⟩ ∈ 𝐴 ∧ ∃𝑦𝑥, 𝑦⟩ ∈ 𝐵))
121, 6, 113imtr4i 292 . 2 (𝑥 ∈ dom (𝐴𝐵) → 𝑥 ∈ (dom 𝐴 ∩ dom 𝐵))
1312ssriv 3983 1 dom (𝐴𝐵) ⊆ (dom 𝐴 ∩ dom 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wa 395  wex 1774  wcel 2099  cin 3944  wss 3945  cop 4631  dom cdm 5673
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-rab 3429  df-v 3472  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4320  df-if 4526  df-sn 4626  df-pr 4628  df-op 4632  df-br 5144  df-dm 5683
This theorem is referenced by:  rnin  6146  psssdm2  18567  hauseqcn  33494
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