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Theorem dmiin 5276
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 4481 . . . 4 𝑥 𝑥𝐴 𝐵
21nfdm 5274 . . 3 𝑥dom 𝑥𝐴 𝐵
32ssiinf 4499 . 2 (dom 𝑥𝐴 𝐵 𝑥𝐴 dom 𝐵 ↔ ∀𝑥𝐴 dom 𝑥𝐴 𝐵 ⊆ dom 𝐵)
4 iinss2 4502 . . 3 (𝑥𝐴 𝑥𝐴 𝐵𝐵)
5 dmss 5231 . . 3 ( 𝑥𝐴 𝐵𝐵 → dom 𝑥𝐴 𝐵 ⊆ dom 𝐵)
64, 5syl 17 . 2 (𝑥𝐴 → dom 𝑥𝐴 𝐵 ⊆ dom 𝐵)
73, 6mprgbir 2910 1 dom 𝑥𝐴 𝐵 𝑥𝐴 dom 𝐵
Colors of variables: wff setvar class
Syntax hints:  wcel 1976  wss 3539   ciin 4450  dom cdm 5027
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rab 2904  df-v 3174  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-iin 4452  df-br 4578  df-dm 5037
This theorem is referenced by: (None)
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