Theorem riin0 5087

Description: Relative intersection of an empty family. (Contributed by Stefan O'Rear, 3-Apr-2015.)

Assertion

Ref	Expression
riin0	⊢ (𝑋 = ∅ → (𝐴 ∩ ∩ 𝑥 ∈ 𝑋 𝑆) = 𝐴)

Distinct variable groups:   𝑥,𝐴   𝑥,𝑋

Allowed substitution hint:   𝑆(𝑥)

Proof of Theorem riin0

Step	Hyp	Ref	Expression
1		iineq1 5014	. . 3 ⊢ (𝑋 = ∅ → ∩ 𝑥 ∈ 𝑋 𝑆 = ∩ 𝑥 ∈ ∅ 𝑆)
2	1	ineq2d 4228	. 2 ⊢ (𝑋 = ∅ → (𝐴 ∩ ∩ 𝑥 ∈ 𝑋 𝑆) = (𝐴 ∩ ∩ 𝑥 ∈ ∅ 𝑆))
3		0iin 5069	. . . 4 ⊢ ∩ 𝑥 ∈ ∅ 𝑆 = V
4	3	ineq2i 4225	. . 3 ⊢ (𝐴 ∩ ∩ 𝑥 ∈ ∅ 𝑆) = (𝐴 ∩ V)
5		inv1 4404	. . 3 ⊢ (𝐴 ∩ V) = 𝐴
6	4, 5	eqtri 2763	. 2 ⊢ (𝐴 ∩ ∩ 𝑥 ∈ ∅ 𝑆) = 𝐴
7	2, 6	eqtrdi 2791	1 ⊢ (𝑋 = ∅ → (𝐴 ∩ ∩ 𝑥 ∈ 𝑋 𝑆) = 𝐴)

Syntax hints:   → wi 4   = wceq 1537  Vcvv 3478   ∩ cin 3962  ∅c0 4339  ∩ ciin 4997

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 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-in 3970  df-ss 3980  df-nul 4340  df-iin 4999

This theorem is referenced by:  riinrab  5089  riiner  8829  mreriincl  17643  riinopn  22930  riincld  23068  fnemeet2  36350  pmapglb2N  39754  pmapglb2xN  39755