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Theorem dmiin 5955
Description: Domain of an intersection. (Contributed by FL, 15-Oct-2012.)
Assertion
Ref Expression
dmiin dom 𝑥𝐴 𝐵 𝑥𝐴 dom 𝐵

Proof of Theorem dmiin
StepHypRef Expression
1 nfii1 5032 . . . 4 𝑥 𝑥𝐴 𝐵
21nfdm 5953 . . 3 𝑥dom 𝑥𝐴 𝐵
32ssiinf 5057 . 2 (dom 𝑥𝐴 𝐵 𝑥𝐴 dom 𝐵 ↔ ∀𝑥𝐴 dom 𝑥𝐴 𝐵 ⊆ dom 𝐵)
4 iinss2 5060 . . 3 (𝑥𝐴 𝑥𝐴 𝐵𝐵)
5 dmss 5905 . . 3 ( 𝑥𝐴 𝐵𝐵 → dom 𝑥𝐴 𝐵 ⊆ dom 𝐵)
64, 5syl 17 . 2 (𝑥𝐴 → dom 𝑥𝐴 𝐵 ⊆ dom 𝐵)
73, 6mprgbir 3065 1 dom 𝑥𝐴 𝐵 𝑥𝐴 dom 𝐵
Colors of variables: wff setvar class
Syntax hints:  wcel 2099  wss 3947   ciin 4997  dom cdm 5678
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ral 3059  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-iin 4999  df-br 5149  df-dm 5688
This theorem is referenced by: (None)
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