Theorem riinm 3938

Description: Relative intersection of an inhabited family. (Contributed by Jim Kingdon, 19-Aug-2018.)

Assertion

Ref	Expression
riinm	⊢ ((∀𝑥 ∈ 𝑋 𝑆 ⊆ 𝐴 ∧ ∃𝑥 𝑥 ∈ 𝑋) → (𝐴 ∩ ∩ 𝑥 ∈ 𝑋 𝑆) = ∩ 𝑥 ∈ 𝑋 𝑆)

Distinct variable groups:   𝑥,𝐴   𝑥,𝑋

Allowed substitution hint:   𝑆(𝑥)

Proof of Theorem riinm

Step	Hyp	Ref	Expression
1		incom 3314	. 2 ⊢ (𝐴 ∩ ∩ 𝑥 ∈ 𝑋 𝑆) = (∩ 𝑥 ∈ 𝑋 𝑆 ∩ 𝐴)
2		r19.2m 3495	. . . . 5 ⊢ ((∃𝑥 𝑥 ∈ 𝑋 ∧ ∀𝑥 ∈ 𝑋 𝑆 ⊆ 𝐴) → ∃𝑥 ∈ 𝑋 𝑆 ⊆ 𝐴)
3	2	ancoms 266	. . . 4 ⊢ ((∀𝑥 ∈ 𝑋 𝑆 ⊆ 𝐴 ∧ ∃𝑥 𝑥 ∈ 𝑋) → ∃𝑥 ∈ 𝑋 𝑆 ⊆ 𝐴)
4		iinss 3917	. . . 4 ⊢ (∃𝑥 ∈ 𝑋 𝑆 ⊆ 𝐴 → ∩ 𝑥 ∈ 𝑋 𝑆 ⊆ 𝐴)
5	3, 4	syl 14	. . 3 ⊢ ((∀𝑥 ∈ 𝑋 𝑆 ⊆ 𝐴 ∧ ∃𝑥 𝑥 ∈ 𝑋) → ∩ 𝑥 ∈ 𝑋 𝑆 ⊆ 𝐴)
6		df-ss 3129	. . 3 ⊢ (∩ 𝑥 ∈ 𝑋 𝑆 ⊆ 𝐴 ↔ (∩ 𝑥 ∈ 𝑋 𝑆 ∩ 𝐴) = ∩ 𝑥 ∈ 𝑋 𝑆)
7	5, 6	sylib 121	. 2 ⊢ ((∀𝑥 ∈ 𝑋 𝑆 ⊆ 𝐴 ∧ ∃𝑥 𝑥 ∈ 𝑋) → (∩ 𝑥 ∈ 𝑋 𝑆 ∩ 𝐴) = ∩ 𝑥 ∈ 𝑋 𝑆)
8	1, 7	syl5eq 2211	1 ⊢ ((∀𝑥 ∈ 𝑋 𝑆 ⊆ 𝐴 ∧ ∃𝑥 𝑥 ∈ 𝑋) → (𝐴 ∩ ∩ 𝑥 ∈ 𝑋 𝑆) = ∩ 𝑥 ∈ 𝑋 𝑆)

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1343  ∃wex 1480   ∈ wcel 2136  ∀wral 2444  ∃wrex 2445   ∩ cin 3115   ⊆ wss 3116  ∩ ciin 3867

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-in 3122  df-ss 3129  df-iin 3869

This theorem is referenced by:  riinerm  6574