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Theorem dmin 5241
Description: The domain of an intersection belong to the intersection of 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 1784 . . 3 (∃𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵) → (∃𝑦𝑥, 𝑦⟩ ∈ 𝐴 ∧ ∃𝑦𝑥, 𝑦⟩ ∈ 𝐵))
2 vex 3175 . . . . 5 𝑥 ∈ V
32eldm2 5231 . . . 4 (𝑥 ∈ dom (𝐴𝐵) ↔ ∃𝑦𝑥, 𝑦⟩ ∈ (𝐴𝐵))
4 elin 3757 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ (𝐴𝐵) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵))
54exbii 1763 . . . 4 (∃𝑦𝑥, 𝑦⟩ ∈ (𝐴𝐵) ↔ ∃𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵))
63, 5bitri 262 . . 3 (𝑥 ∈ dom (𝐴𝐵) ↔ ∃𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵))
7 elin 3757 . . . 4 (𝑥 ∈ (dom 𝐴 ∩ dom 𝐵) ↔ (𝑥 ∈ dom 𝐴𝑥 ∈ dom 𝐵))
82eldm2 5231 . . . . 5 (𝑥 ∈ dom 𝐴 ↔ ∃𝑦𝑥, 𝑦⟩ ∈ 𝐴)
92eldm2 5231 . . . . 5 (𝑥 ∈ dom 𝐵 ↔ ∃𝑦𝑥, 𝑦⟩ ∈ 𝐵)
108, 9anbi12i 728 . . . 4 ((𝑥 ∈ dom 𝐴𝑥 ∈ dom 𝐵) ↔ (∃𝑦𝑥, 𝑦⟩ ∈ 𝐴 ∧ ∃𝑦𝑥, 𝑦⟩ ∈ 𝐵))
117, 10bitri 262 . . 3 (𝑥 ∈ (dom 𝐴 ∩ dom 𝐵) ↔ (∃𝑦𝑥, 𝑦⟩ ∈ 𝐴 ∧ ∃𝑦𝑥, 𝑦⟩ ∈ 𝐵))
121, 6, 113imtr4i 279 . 2 (𝑥 ∈ dom (𝐴𝐵) → 𝑥 ∈ (dom 𝐴 ∩ dom 𝐵))
1312ssriv 3571 1 dom (𝐴𝐵) ⊆ (dom 𝐴 ∩ dom 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wa 382  wex 1694  wcel 1976  cin 3538  wss 3539  cop 4130  dom cdm 5028
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-rab 2904  df-v 3174  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-br 4578  df-dm 5038
This theorem is referenced by:  rnin  5447  psssdm2  16984  hauseqcn  29075
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