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Theorem dmiin 5909
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 4990 . . . 4 𝑥 𝑥𝐴 𝐵
21nfdm 5907 . . 3 𝑥dom 𝑥𝐴 𝐵
32ssiinf 5015 . 2 (dom 𝑥𝐴 𝐵 𝑥𝐴 dom 𝐵 ↔ ∀𝑥𝐴 dom 𝑥𝐴 𝐵 ⊆ dom 𝐵)
4 iinss2 5018 . . 3 (𝑥𝐴 𝑥𝐴 𝐵𝐵)
5 dmss 5859 . . 3 ( 𝑥𝐴 𝐵𝐵 → dom 𝑥𝐴 𝐵 ⊆ dom 𝐵)
64, 5syl 17 . 2 (𝑥𝐴 → dom 𝑥𝐴 𝐵 ⊆ dom 𝐵)
73, 6mprgbir 3068 1 dom 𝑥𝐴 𝐵 𝑥𝐴 dom 𝐵
Colors of variables: wff setvar class
Syntax hints:  wcel 2107  wss 3911   ciin 4956  dom cdm 5634
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3062  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-iin 4958  df-br 5107  df-dm 5644
This theorem is referenced by: (None)
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