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Theorem dmin 4886
 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

Proof of Theorem dmin
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 19.40 1596 . . 3
2 vex 2791 . . . . 5
32eldm2 4877 . . . 4
4 elin 3358 . . . . 5
54exbii 1569 . . . 4
63, 5bitri 240 . . 3
7 elin 3358 . . . 4
82eldm2 4877 . . . . 5
92eldm2 4877 . . . . 5
108, 9anbi12i 678 . . . 4
117, 10bitri 240 . . 3
121, 6, 113imtr4i 257 . 2
1312ssriv 3184 1
 Colors of variables: wff set class Syntax hints:   wa 358  wex 1528   wcel 1684   cin 3151   wss 3152  cop 3643   cdm 4689 This theorem is referenced by:  rnin  5090  psssdm2  14324  domintrefb  25063 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-dm 4699
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