Theorem riinn0 5053

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 4170	. 2 ⊢ (𝐴 ∩ ∩ 𝑥 ∈ 𝑋 𝑆) = (∩ 𝑥 ∈ 𝑋 𝑆 ∩ 𝐴)
2		r19.2z 4465	. . . . 5 ⊢ ((𝑋 ≠ ∅ ∧ ∀𝑥 ∈ 𝑋 𝑆 ⊆ 𝐴) → ∃𝑥 ∈ 𝑋 𝑆 ⊆ 𝐴)
3	2	ancoms 463	. . . 4 ⊢ ((∀𝑥 ∈ 𝑋 𝑆 ⊆ 𝐴 ∧ 𝑋 ≠ ∅) → ∃𝑥 ∈ 𝑋 𝑆 ⊆ 𝐴)
4		iinss 5025	. . . 4 ⊢ (∃𝑥 ∈ 𝑋 𝑆 ⊆ 𝐴 → ∩ 𝑥 ∈ 𝑋 𝑆 ⊆ 𝐴)
5	3, 4	syl 18	. . 3 ⊢ ((∀𝑥 ∈ 𝑋 𝑆 ⊆ 𝐴 ∧ 𝑋 ≠ ∅) → ∩ 𝑥 ∈ 𝑋 𝑆 ⊆ 𝐴)
6		dfss2 3931	. . 3 ⊢ (∩ 𝑥 ∈ 𝑋 𝑆 ⊆ 𝐴 ↔ (∩ 𝑥 ∈ 𝑋 𝑆 ∩ 𝐴) = ∩ 𝑥 ∈ 𝑋 𝑆)
7	5, 6	sylib 221	. 2 ⊢ ((∀𝑥 ∈ 𝑋 𝑆 ⊆ 𝐴 ∧ 𝑋 ≠ ∅) → (∩ 𝑥 ∈ 𝑋 𝑆 ∩ 𝐴) = ∩ 𝑥 ∈ 𝑋 𝑆)
8	1, 7	eqtrid 2816	1 ⊢ ((∀𝑥 ∈ 𝑋 𝑆 ⊆ 𝐴 ∧ 𝑋 ≠ ∅) → (𝐴 ∩ ∩ 𝑥 ∈ 𝑋 𝑆) = ∩ 𝑥 ∈ 𝑋 𝑆)

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1567   ≠ wne 2964  ∀wral 3085  ∃wrex 3095   ∩ cin 3912   ⊆ wss 3913  ∅c0 4294  ∩ ciin 4961

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-in 3920  df-ss 3930  df-nul 4295  df-iin 4963

This theorem is referenced by:  riinrab  5054  riiner  8788  mreriincl  17650  riinopn  23034  alexsublem  24170  fnemeet1  36766