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Theorem dmiin 5793
 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 4919 . . . 4 𝑥 𝑥𝐴 𝐵
21nfdm 5791 . . 3 𝑥dom 𝑥𝐴 𝐵
32ssiinf 4944 . 2 (dom 𝑥𝐴 𝐵 𝑥𝐴 dom 𝐵 ↔ ∀𝑥𝐴 dom 𝑥𝐴 𝐵 ⊆ dom 𝐵)
4 iinss2 4947 . . 3 (𝑥𝐴 𝑥𝐴 𝐵𝐵)
5 dmss 5739 . . 3 ( 𝑥𝐴 𝐵𝐵 → dom 𝑥𝐴 𝐵 ⊆ dom 𝐵)
64, 5syl 17 . 2 (𝑥𝐴 → dom 𝑥𝐴 𝐵 ⊆ dom 𝐵)
73, 6mprgbir 3124 1 dom 𝑥𝐴 𝐵 𝑥𝐴 dom 𝐵
 Colors of variables: wff setvar class Syntax hints:   ∈ wcel 2112   ⊆ wss 3884  ∩ ciin 4885  dom cdm 5523 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-v 3446  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-iin 4887  df-br 5034  df-dm 5533 This theorem is referenced by: (None)
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