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Theorem dmin 5748
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 1887 . . 3 (∃𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵) → (∃𝑦𝑥, 𝑦⟩ ∈ 𝐴 ∧ ∃𝑦𝑥, 𝑦⟩ ∈ 𝐵))
2 vex 3447 . . . . 5 𝑥 ∈ V
32eldm2 5738 . . . 4 (𝑥 ∈ dom (𝐴𝐵) ↔ ∃𝑦𝑥, 𝑦⟩ ∈ (𝐴𝐵))
4 elin 3900 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ (𝐴𝐵) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵))
54exbii 1849 . . . 4 (∃𝑦𝑥, 𝑦⟩ ∈ (𝐴𝐵) ↔ ∃𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵))
63, 5bitri 278 . . 3 (𝑥 ∈ dom (𝐴𝐵) ↔ ∃𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵))
7 elin 3900 . . . 4 (𝑥 ∈ (dom 𝐴 ∩ dom 𝐵) ↔ (𝑥 ∈ dom 𝐴𝑥 ∈ dom 𝐵))
82eldm2 5738 . . . . 5 (𝑥 ∈ dom 𝐴 ↔ ∃𝑦𝑥, 𝑦⟩ ∈ 𝐴)
92eldm2 5738 . . . . 5 (𝑥 ∈ dom 𝐵 ↔ ∃𝑦𝑥, 𝑦⟩ ∈ 𝐵)
108, 9anbi12i 629 . . . 4 ((𝑥 ∈ dom 𝐴𝑥 ∈ dom 𝐵) ↔ (∃𝑦𝑥, 𝑦⟩ ∈ 𝐴 ∧ ∃𝑦𝑥, 𝑦⟩ ∈ 𝐵))
117, 10bitri 278 . . 3 (𝑥 ∈ (dom 𝐴 ∩ dom 𝐵) ↔ (∃𝑦𝑥, 𝑦⟩ ∈ 𝐴 ∧ ∃𝑦𝑥, 𝑦⟩ ∈ 𝐵))
121, 6, 113imtr4i 295 . 2 (𝑥 ∈ dom (𝐴𝐵) → 𝑥 ∈ (dom 𝐴 ∩ dom 𝐵))
1312ssriv 3922 1 dom (𝐴𝐵) ⊆ (dom 𝐴 ∩ dom 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wa 399  wex 1781  wcel 2112  cin 3883  wss 3884  cop 4534  dom cdm 5523
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-v 3446  df-un 3889  df-in 3891  df-ss 3901  df-sn 4529  df-pr 4531  df-op 4535  df-br 5034  df-dm 5533
This theorem is referenced by:  rnin  5976  psssdm2  17820  hauseqcn  31249
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