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Theorem riin0 5007
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 4938 . . 3 (𝑋 = ∅ → 𝑥𝑋 𝑆 = 𝑥 ∈ ∅ 𝑆)
21ineq2d 4144 . 2 (𝑋 = ∅ → (𝐴 𝑥𝑋 𝑆) = (𝐴 𝑥 ∈ ∅ 𝑆))
3 0iin 4989 . . . 4 𝑥 ∈ ∅ 𝑆 = V
43ineq2i 4141 . . 3 (𝐴 𝑥 ∈ ∅ 𝑆) = (𝐴 ∩ V)
5 inv1 4326 . . 3 (𝐴 ∩ V) = 𝐴
64, 5eqtri 2767 . 2 (𝐴 𝑥 ∈ ∅ 𝑆) = 𝐴
72, 6eqtrdi 2796 1 (𝑋 = ∅ → (𝐴 𝑥𝑋 𝑆) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1543  Vcvv 3423  cin 3882  c0 4254   ciin 4922
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2114  ax-9 2122  ax-ext 2710
This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2073  df-clab 2717  df-cleq 2731  df-clel 2818  df-ral 3069  df-rab 3073  df-v 3425  df-dif 3886  df-in 3890  df-ss 3900  df-nul 4255  df-iin 4924
This theorem is referenced by:  riinrab  5009  riiner  8496  mreriincl  17134  riinopn  21837  riincld  21973  fnemeet2  34327  pmapglb2N  37559  pmapglb2xN  37560
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