Theorem rintm 3744

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 3129	. 2 ⊢ (𝐴 ∩ ∩ 𝑋) = (∩ 𝑋 ∩ 𝐴)
2		intssuni2m 3639	. . . 4 ⊢ ((𝑋 ⊆ 𝒫 𝐴 ∧ ∃𝑥 𝑥 ∈ 𝑋) → ∩ 𝑋 ⊆ ∪ 𝒫 𝐴)
3		ssid 2964	. . . . 5 ⊢ 𝒫 𝐴 ⊆ 𝒫 𝐴
4		sspwuni 3739	. . . . 5 ⊢ (𝒫 𝐴 ⊆ 𝒫 𝐴 ↔ ∪ 𝒫 𝐴 ⊆ 𝐴)
5	3, 4	mpbi 133	. . . 4 ⊢ ∪ 𝒫 𝐴 ⊆ 𝐴
6	2, 5	syl6ss 2957	. . 3 ⊢ ((𝑋 ⊆ 𝒫 𝐴 ∧ ∃𝑥 𝑥 ∈ 𝑋) → ∩ 𝑋 ⊆ 𝐴)
7		df-ss 2931	. . 3 ⊢ (∩ 𝑋 ⊆ 𝐴 ↔ (∩ 𝑋 ∩ 𝐴) = ∩ 𝑋)
8	6, 7	sylib 127	. 2 ⊢ ((𝑋 ⊆ 𝒫 𝐴 ∧ ∃𝑥 𝑥 ∈ 𝑋) → (∩ 𝑋 ∩ 𝐴) = ∩ 𝑋)
9	1, 8	syl5eq 2084	1 ⊢ ((𝑋 ⊆ 𝒫 𝐴 ∧ ∃𝑥 𝑥 ∈ 𝑋) → (𝐴 ∩ ∩ 𝑋) = ∩ 𝑋)

Syntax hints:   → wi 4   ∧ wa 97   = wceq 1243  ∃wex 1381   ∈ wcel 1393   ∩ cin 2916   ⊆ wss 2917  𝒫 cpw 3359  ∪ cuni 3580  ∩ cint 3615

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-in 2924  df-ss 2931  df-pw 3361  df-uni 3581  df-int 3616

This theorem is referenced by: (None)