Theorem riinn0 5012

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

Assertion

Ref	Expression
riinn0	⊢ ((∀𝑥 ∈ 𝑋 𝑆 ⊆ 𝐴 ∧ 𝑋 ≠ ∅) → (𝐴 ∩ ∩ 𝑥 ∈ 𝑋 𝑆) = ∩ 𝑥 ∈ 𝑋 𝑆)

Distinct variable groups:   𝑥,𝐴   𝑥,𝑋

Allowed substitution hint:   𝑆(𝑥)

Proof of Theorem riinn0

Step	Hyp	Ref	Expression
1		incom 4135	. 2 ⊢ (𝐴 ∩ ∩ 𝑥 ∈ 𝑋 𝑆) = (∩ 𝑥 ∈ 𝑋 𝑆 ∩ 𝐴)
2		r19.2z 4425	. . . . 5 ⊢ ((𝑋 ≠ ∅ ∧ ∀𝑥 ∈ 𝑋 𝑆 ⊆ 𝐴) → ∃𝑥 ∈ 𝑋 𝑆 ⊆ 𝐴)
3	2	ancoms 459	. . . 4 ⊢ ((∀𝑥 ∈ 𝑋 𝑆 ⊆ 𝐴 ∧ 𝑋 ≠ ∅) → ∃𝑥 ∈ 𝑋 𝑆 ⊆ 𝐴)
4		iinss 4986	. . . 4 ⊢ (∃𝑥 ∈ 𝑋 𝑆 ⊆ 𝐴 → ∩ 𝑥 ∈ 𝑋 𝑆 ⊆ 𝐴)
5	3, 4	syl 17	. . 3 ⊢ ((∀𝑥 ∈ 𝑋 𝑆 ⊆ 𝐴 ∧ 𝑋 ≠ ∅) → ∩ 𝑥 ∈ 𝑋 𝑆 ⊆ 𝐴)
6		df-ss 3904	. . 3 ⊢ (∩ 𝑥 ∈ 𝑋 𝑆 ⊆ 𝐴 ↔ (∩ 𝑥 ∈ 𝑋 𝑆 ∩ 𝐴) = ∩ 𝑥 ∈ 𝑋 𝑆)
7	5, 6	sylib 217	. 2 ⊢ ((∀𝑥 ∈ 𝑋 𝑆 ⊆ 𝐴 ∧ 𝑋 ≠ ∅) → (∩ 𝑥 ∈ 𝑋 𝑆 ∩ 𝐴) = ∩ 𝑥 ∈ 𝑋 𝑆)
8	1, 7	eqtrid 2790	1 ⊢ ((∀𝑥 ∈ 𝑋 𝑆 ⊆ 𝐴 ∧ 𝑋 ≠ ∅) → (𝐴 ∩ ∩ 𝑥 ∈ 𝑋 𝑆) = ∩ 𝑥 ∈ 𝑋 𝑆)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539   ≠ wne 2943  ∀wral 3064  ∃wrex 3065   ∩ cin 3886   ⊆ wss 3887  ∅c0 4256  ∩ ciin 4925

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-in 3894  df-ss 3904  df-nul 4257  df-iin 4927

This theorem is referenced by:  riinrab  5013  riiner  8579  mreriincl  17307  riinopn  22057  alexsublem  23195  fnemeet1  34555