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Theorem riin0 4996
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 4928 . . 3 (𝑋 = ∅ → 𝑥𝑋 𝑆 = 𝑥 ∈ ∅ 𝑆)
21ineq2d 4188 . 2 (𝑋 = ∅ → (𝐴 𝑥𝑋 𝑆) = (𝐴 𝑥 ∈ ∅ 𝑆))
3 0iin 4979 . . . 4 𝑥 ∈ ∅ 𝑆 = V
43ineq2i 4185 . . 3 (𝐴 𝑥 ∈ ∅ 𝑆) = (𝐴 ∩ V)
5 inv1 4347 . . 3 (𝐴 ∩ V) = 𝐴
64, 5eqtri 2844 . 2 (𝐴 𝑥 ∈ ∅ 𝑆) = 𝐴
72, 6syl6eq 2872 1 (𝑋 = ∅ → (𝐴 𝑥𝑋 𝑆) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  Vcvv 3494  cin 3934  c0 4290   ciin 4912
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rab 3147  df-v 3496  df-dif 3938  df-in 3942  df-ss 3951  df-nul 4291  df-iin 4914
This theorem is referenced by:  riinrab  4998  riiner  8364  mreriincl  16863  riinopn  21510  riincld  21646  fnemeet2  33710  pmapglb2N  36901  pmapglb2xN  36902
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