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Theorem dmin 4758
 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 1611 . . 3 (∃𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵) → (∃𝑦𝑥, 𝑦⟩ ∈ 𝐴 ∧ ∃𝑦𝑥, 𝑦⟩ ∈ 𝐵))
2 vex 2693 . . . . 5 𝑥 ∈ V
32eldm2 4748 . . . 4 (𝑥 ∈ dom (𝐴𝐵) ↔ ∃𝑦𝑥, 𝑦⟩ ∈ (𝐴𝐵))
4 elin 3265 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ (𝐴𝐵) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵))
54exbii 1585 . . . 4 (∃𝑦𝑥, 𝑦⟩ ∈ (𝐴𝐵) ↔ ∃𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵))
63, 5bitri 183 . . 3 (𝑥 ∈ dom (𝐴𝐵) ↔ ∃𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵))
7 elin 3265 . . . 4 (𝑥 ∈ (dom 𝐴 ∩ dom 𝐵) ↔ (𝑥 ∈ dom 𝐴𝑥 ∈ dom 𝐵))
82eldm2 4748 . . . . 5 (𝑥 ∈ dom 𝐴 ↔ ∃𝑦𝑥, 𝑦⟩ ∈ 𝐴)
92eldm2 4748 . . . . 5 (𝑥 ∈ dom 𝐵 ↔ ∃𝑦𝑥, 𝑦⟩ ∈ 𝐵)
108, 9anbi12i 456 . . . 4 ((𝑥 ∈ dom 𝐴𝑥 ∈ dom 𝐵) ↔ (∃𝑦𝑥, 𝑦⟩ ∈ 𝐴 ∧ ∃𝑦𝑥, 𝑦⟩ ∈ 𝐵))
117, 10bitri 183 . . 3 (𝑥 ∈ (dom 𝐴 ∩ dom 𝐵) ↔ (∃𝑦𝑥, 𝑦⟩ ∈ 𝐴 ∧ ∃𝑦𝑥, 𝑦⟩ ∈ 𝐵))
121, 6, 113imtr4i 200 . 2 (𝑥 ∈ dom (𝐴𝐵) → 𝑥 ∈ (dom 𝐴 ∩ dom 𝐵))
1312ssriv 3107 1 dom (𝐴𝐵) ⊆ (dom 𝐴 ∩ dom 𝐵)
 Colors of variables: wff set class Syntax hints:   ∧ wa 103  ∃wex 1469   ∈ wcel 1481   ∩ cin 3076   ⊆ wss 3077  ⟨cop 3536  dom cdm 4550 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2692  df-un 3081  df-in 3083  df-ss 3090  df-sn 3539  df-pr 3540  df-op 3542  df-br 3939  df-dm 4560 This theorem is referenced by:  rnin  4959
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