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Theorem dmin 4486
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 1519 . . 3 (y(⟨x, y A x, y B) → (yx, y A yx, y B))
2 vex 2554 . . . . 5 x V
32eldm2 4476 . . . 4 (x dom (AB) ↔ yx, y (AB))
4 elin 3120 . . . . 5 (⟨x, y (AB) ↔ (⟨x, y A x, y B))
54exbii 1493 . . . 4 (yx, y (AB) ↔ y(⟨x, y A x, y B))
63, 5bitri 173 . . 3 (x dom (AB) ↔ y(⟨x, y A x, y B))
7 elin 3120 . . . 4 (x (dom A ∩ dom B) ↔ (x dom A x dom B))
82eldm2 4476 . . . . 5 (x dom Ayx, y A)
92eldm2 4476 . . . . 5 (x dom Byx, y B)
108, 9anbi12i 433 . . . 4 ((x dom A x dom B) ↔ (yx, y A yx, y B))
117, 10bitri 173 . . 3 (x (dom A ∩ dom B) ↔ (yx, y A yx, y B))
121, 6, 113imtr4i 190 . 2 (x dom (AB) → x (dom A ∩ dom B))
1312ssriv 2943 1 dom (AB) ⊆ (dom A ∩ dom B)
Colors of variables: wff set class
Syntax hints:   wa 97  wex 1378   wcel 1390  cin 2910  wss 2911  cop 3370  dom cdm 4288
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-dm 4298
This theorem is referenced by:  rnin  4676
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