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Theorem dmin 4570
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 1538 . . 3 (∃𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵) → (∃𝑦𝑥, 𝑦⟩ ∈ 𝐴 ∧ ∃𝑦𝑥, 𝑦⟩ ∈ 𝐵))
2 vex 2577 . . . . 5 𝑥 ∈ V
32eldm2 4560 . . . 4 (𝑥 ∈ dom (𝐴𝐵) ↔ ∃𝑦𝑥, 𝑦⟩ ∈ (𝐴𝐵))
4 elin 3153 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ (𝐴𝐵) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵))
54exbii 1512 . . . 4 (∃𝑦𝑥, 𝑦⟩ ∈ (𝐴𝐵) ↔ ∃𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵))
63, 5bitri 177 . . 3 (𝑥 ∈ dom (𝐴𝐵) ↔ ∃𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵))
7 elin 3153 . . . 4 (𝑥 ∈ (dom 𝐴 ∩ dom 𝐵) ↔ (𝑥 ∈ dom 𝐴𝑥 ∈ dom 𝐵))
82eldm2 4560 . . . . 5 (𝑥 ∈ dom 𝐴 ↔ ∃𝑦𝑥, 𝑦⟩ ∈ 𝐴)
92eldm2 4560 . . . . 5 (𝑥 ∈ dom 𝐵 ↔ ∃𝑦𝑥, 𝑦⟩ ∈ 𝐵)
108, 9anbi12i 441 . . . 4 ((𝑥 ∈ dom 𝐴𝑥 ∈ dom 𝐵) ↔ (∃𝑦𝑥, 𝑦⟩ ∈ 𝐴 ∧ ∃𝑦𝑥, 𝑦⟩ ∈ 𝐵))
117, 10bitri 177 . . 3 (𝑥 ∈ (dom 𝐴 ∩ dom 𝐵) ↔ (∃𝑦𝑥, 𝑦⟩ ∈ 𝐴 ∧ ∃𝑦𝑥, 𝑦⟩ ∈ 𝐵))
121, 6, 113imtr4i 194 . 2 (𝑥 ∈ dom (𝐴𝐵) → 𝑥 ∈ (dom 𝐴 ∩ dom 𝐵))
1312ssriv 2976 1 dom (𝐴𝐵) ⊆ (dom 𝐴 ∩ dom 𝐵)
Colors of variables: wff set class
Syntax hints:  wa 101  wex 1397  wcel 1409  cin 2943  wss 2944  cop 3405  dom cdm 4372
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-un 2949  df-in 2951  df-ss 2958  df-sn 3408  df-pr 3409  df-op 3411  df-br 3792  df-dm 4382
This theorem is referenced by:  rnin  4760
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