Theorem riin0 5041

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 4968	. . 3 ⊢ (𝑋 = ∅ → ∩ 𝑥 ∈ 𝑋 𝑆 = ∩ 𝑥 ∈ ∅ 𝑆)
2	1	ineq2d 4117	. 2 ⊢ (𝑋 = ∅ → (𝐴 ∩ ∩ 𝑥 ∈ 𝑋 𝑆) = (𝐴 ∩ ∩ 𝑥 ∈ ∅ 𝑆))
3		0iin 5021	. . . 4 ⊢ ∩ 𝑥 ∈ ∅ 𝑆 = V
4	3	ineq2i 4114	. . 3 ⊢ (𝐴 ∩ ∩ 𝑥 ∈ ∅ 𝑆) = (𝐴 ∩ V)
5		inv1 4347	. . 3 ⊢ (𝐴 ∩ V) = 𝐴
6	4, 5	eqtri 2783	. 2 ⊢ (𝐴 ∩ ∩ 𝑥 ∈ ∅ 𝑆) = 𝐴
7	2, 6	eqtrdi 2811	1 ⊢ (𝑋 = ∅ → (𝐴 ∩ ∩ 𝑥 ∈ 𝑋 𝑆) = 𝐴)

This proof depends on syntax axioms:   → wi 4   = wceq 1569  Vcvv 3450   ∩ cin 4089  ∅c0 4278  ∩ ciin 4951

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-nul 4279  df-iin 4953

This theorem is used by:  riinrab  5043  riiner  8811  mreriincl  17768  riinopn  23226  riincld  23362  fnemeet2  37155  pmapglb2N  40829  pmapglb2xN  40830