Theorem rintm 3965

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

Assertion

Ref	Expression
rintm	⊢ ((𝑋 ⊆ 𝒫 𝐴 ∧ ∃𝑥 𝑥 ∈ 𝑋) → (𝐴 ∩ ∩ 𝑋) = ∩ 𝑋)

Distinct variable group:   𝑥,𝑋

Allowed substitution hint:   𝐴(𝑥)

Proof of Theorem rintm

Step	Hyp	Ref	Expression
1		incom 3319	. 2 ⊢ (𝐴 ∩ ∩ 𝑋) = (∩ 𝑋 ∩ 𝐴)
2		intssuni2m 3855	. . . 4 ⊢ ((𝑋 ⊆ 𝒫 𝐴 ∧ ∃𝑥 𝑥 ∈ 𝑋) → ∩ 𝑋 ⊆ ∪ 𝒫 𝐴)
3		ssid 3167	. . . . 5 ⊢ 𝒫 𝐴 ⊆ 𝒫 𝐴
4		sspwuni 3957	. . . . 5 ⊢ (𝒫 𝐴 ⊆ 𝒫 𝐴 ↔ ∪ 𝒫 𝐴 ⊆ 𝐴)
5	3, 4	mpbi 144	. . . 4 ⊢ ∪ 𝒫 𝐴 ⊆ 𝐴
6	2, 5	sstrdi 3159	. . 3 ⊢ ((𝑋 ⊆ 𝒫 𝐴 ∧ ∃𝑥 𝑥 ∈ 𝑋) → ∩ 𝑋 ⊆ 𝐴)
7		df-ss 3134	. . 3 ⊢ (∩ 𝑋 ⊆ 𝐴 ↔ (∩ 𝑋 ∩ 𝐴) = ∩ 𝑋)
8	6, 7	sylib 121	. 2 ⊢ ((𝑋 ⊆ 𝒫 𝐴 ∧ ∃𝑥 𝑥 ∈ 𝑋) → (∩ 𝑋 ∩ 𝐴) = ∩ 𝑋)
9	1, 8	eqtrid 2215	1 ⊢ ((𝑋 ⊆ 𝒫 𝐴 ∧ ∃𝑥 𝑥 ∈ 𝑋) → (𝐴 ∩ ∩ 𝑋) = ∩ 𝑋)

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1348  ∃wex 1485   ∈ wcel 2141   ∩ cin 3120   ⊆ wss 3121  𝒫 cpw 3566  ∪ cuni 3796  ∩ cint 3831

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-in 3127  df-ss 3134  df-pw 3568  df-uni 3797  df-int 3832

This theorem is referenced by: (None)