Theorem riin0 5014

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 4942	. . 3 ⊢ (𝑋 = ∅ → ∩ 𝑥 ∈ 𝑋 𝑆 = ∩ 𝑥 ∈ ∅ 𝑆)
2	1	ineq2d 4152	. 2 ⊢ (𝑋 = ∅ → (𝐴 ∩ ∩ 𝑥 ∈ 𝑋 𝑆) = (𝐴 ∩ ∩ 𝑥 ∈ ∅ 𝑆))
3		0iin 4996	. . . 4 ⊢ ∩ 𝑥 ∈ ∅ 𝑆 = V
4	3	ineq2i 4149	. . 3 ⊢ (𝐴 ∩ ∩ 𝑥 ∈ ∅ 𝑆) = (𝐴 ∩ V)
5		inv1 4329	. . 3 ⊢ (𝐴 ∩ V) = 𝐴
6	4, 5	eqtri 2764	. 2 ⊢ (𝐴 ∩ ∩ 𝑥 ∈ ∅ 𝑆) = 𝐴
7	2, 6	eqtrdi 2792	1 ⊢ (𝑋 = ∅ → (𝐴 ∩ ∩ 𝑥 ∈ 𝑋 𝑆) = 𝐴)

Syntax hints:   → wi 4   = wceq 1548  Vcvv 3433   ∩ cin 3884  ∅c0 4264  ∩ ciin 4925

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 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713

This theorem depends on definitions:  df-bi 209  df-an 398  df-tru 1551  df-fal 1561  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-ral 3056  df-rex 3066  df-rab 3394  df-v 3435  df-dif 3888  df-in 3892  df-ss 3902  df-nul 4265  df-iin 4927

This theorem is referenced by:  riinrab  5016  riiner  8731  mreriincl  17555  riinopn  22895  riincld  23031  fnemeet2  36610  pmapglb2N  40278  pmapglb2xN  40279