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Theorem riin0 3957
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 3900 . . 3 (𝑋 = ∅ → 𝑥𝑋 𝑆 = 𝑥 ∈ ∅ 𝑆)
21ineq2d 3336 . 2 (𝑋 = ∅ → (𝐴 𝑥𝑋 𝑆) = (𝐴 𝑥 ∈ ∅ 𝑆))
3 0iin 3944 . . . 4 𝑥 ∈ ∅ 𝑆 = V
43ineq2i 3333 . . 3 (𝐴 𝑥 ∈ ∅ 𝑆) = (𝐴 ∩ V)
5 inv1 3459 . . 3 (𝐴 ∩ V) = 𝐴
64, 5eqtri 2198 . 2 (𝐴 𝑥 ∈ ∅ 𝑆) = 𝐴
72, 6eqtrdi 2226 1 (𝑋 = ∅ → (𝐴 𝑥𝑋 𝑆) = 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1353  Vcvv 2737  cin 3128  c0 3422   ciin 3887
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-v 2739  df-dif 3131  df-in 3135  df-ss 3142  df-nul 3423  df-iin 3889
This theorem is referenced by: (None)
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