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Theorem riin0 4524
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 4465 . . 3 (𝑋 = ∅ → 𝑥𝑋 𝑆 = 𝑥 ∈ ∅ 𝑆)
21ineq2d 3775 . 2 (𝑋 = ∅ → (𝐴 𝑥𝑋 𝑆) = (𝐴 𝑥 ∈ ∅ 𝑆))
3 0iin 4508 . . . 4 𝑥 ∈ ∅ 𝑆 = V
43ineq2i 3772 . . 3 (𝐴 𝑥 ∈ ∅ 𝑆) = (𝐴 ∩ V)
5 inv1 3921 . . 3 (𝐴 ∩ V) = 𝐴
64, 5eqtri 2631 . 2 (𝐴 𝑥 ∈ ∅ 𝑆) = 𝐴
72, 6syl6eq 2659 1 (𝑋 = ∅ → (𝐴 𝑥𝑋 𝑆) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1474  Vcvv 3172  cin 3538  c0 3873   ciin 4450
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-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-v 3174  df-dif 3542  df-in 3546  df-ss 3553  df-nul 3874  df-iin 4452
This theorem is referenced by:  riinrab  4526  riiner  7684  mreriincl  16027  riinopn  20480  riincld  20600  fnemeet2  31338  pmapglb2N  33871  pmapglb2xN  33872
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