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Theorem dmiin 5290
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 4487 . . . 4 𝑥 𝑥𝐴 𝐵
21nfdm 5288 . . 3 𝑥dom 𝑥𝐴 𝐵
32ssiinf 4505 . 2 (dom 𝑥𝐴 𝐵 𝑥𝐴 dom 𝐵 ↔ ∀𝑥𝐴 dom 𝑥𝐴 𝐵 ⊆ dom 𝐵)
4 iinss2 4508 . . 3 (𝑥𝐴 𝑥𝐴 𝐵𝐵)
5 dmss 5245 . . 3 ( 𝑥𝐴 𝐵𝐵 → dom 𝑥𝐴 𝐵 ⊆ dom 𝐵)
64, 5syl 17 . 2 (𝑥𝐴 → dom 𝑥𝐴 𝐵 ⊆ dom 𝐵)
73, 6mprgbir 2911 1 dom 𝑥𝐴 𝐵 𝑥𝐴 dom 𝐵
Colors of variables: wff setvar class
Syntax hints:  wcel 1977  wss 3540   ciin 4456  dom cdm 5038
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-iin 4458  df-br 4584  df-dm 5048
This theorem is referenced by: (None)
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