MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  riin0 Structured version   Visualization version   GIF version

Theorem riin0 4788
Description: Relative intersection of an empty family. (Contributed by Stefan O'Rear, 3-Apr-2015.)
Assertion
Ref Expression
riin0 (𝑋 = ∅ → (𝐴 𝑥𝑋 𝑆) = 𝐴)
Distinct variable groups:   𝑥,𝐴   𝑥,𝑋
Allowed substitution hint:   𝑆(𝑥)

Proof of Theorem riin0
StepHypRef Expression
1 iineq1 4729 . . 3 (𝑋 = ∅ → 𝑥𝑋 𝑆 = 𝑥 ∈ ∅ 𝑆)
21ineq2d 4016 . 2 (𝑋 = ∅ → (𝐴 𝑥𝑋 𝑆) = (𝐴 𝑥 ∈ ∅ 𝑆))
3 0iin 4772 . . . 4 𝑥 ∈ ∅ 𝑆 = V
43ineq2i 4013 . . 3 (𝐴 𝑥 ∈ ∅ 𝑆) = (𝐴 ∩ V)
5 inv1 4170 . . 3 (𝐴 ∩ V) = 𝐴
64, 5eqtri 2825 . 2 (𝐴 𝑥 ∈ ∅ 𝑆) = 𝐴
72, 6syl6eq 2853 1 (𝑋 = ∅ → (𝐴 𝑥𝑋 𝑆) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1653  Vcvv 3389  cin 3772  c0 4119   ciin 4715
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-ext 2781
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2790  df-cleq 2796  df-clel 2799  df-nfc 2934  df-ral 3098  df-v 3391  df-dif 3776  df-in 3780  df-ss 3787  df-nul 4120  df-iin 4717
This theorem is referenced by:  riinrab  4790  riiner  8062  mreriincl  16577  riinopn  21045  riincld  21181  fnemeet2  32878  pmapglb2N  35796  pmapglb2xN  35797
  Copyright terms: Public domain W3C validator