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Theorem riin0 5086
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 5014 . . 3 (𝑋 = ∅ → 𝑥𝑋 𝑆 = 𝑥 ∈ ∅ 𝑆)
21ineq2d 4210 . 2 (𝑋 = ∅ → (𝐴 𝑥𝑋 𝑆) = (𝐴 𝑥 ∈ ∅ 𝑆))
3 0iin 5068 . . . 4 𝑥 ∈ ∅ 𝑆 = V
43ineq2i 4207 . . 3 (𝐴 𝑥 ∈ ∅ 𝑆) = (𝐴 ∩ V)
5 inv1 4396 . . 3 (𝐴 ∩ V) = 𝐴
64, 5eqtri 2753 . 2 (𝐴 𝑥 ∈ ∅ 𝑆) = 𝐴
72, 6eqtrdi 2781 1 (𝑋 = ∅ → (𝐴 𝑥𝑋 𝑆) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  Vcvv 3461  cin 3943  c0 4322   ciin 4998
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696
This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-in 3951  df-ss 3961  df-nul 4323  df-iin 5000
This theorem is referenced by:  riinrab  5088  riiner  8809  mreriincl  17581  riinopn  22854  riincld  22992  fnemeet2  35982  pmapglb2N  39374  pmapglb2xN  39375
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