Theorem riinm 3729

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 3129	. 2 ⊢ (𝐴 ∩ ∩ 𝑥 ∈ 𝑋 𝑆) = (∩ 𝑥 ∈ 𝑋 𝑆 ∩ 𝐴)
2		r19.2m 3309	. . . . 5 ⊢ ((∃𝑥 𝑥 ∈ 𝑋 ∧ ∀𝑥 ∈ 𝑋 𝑆 ⊆ 𝐴) → ∃𝑥 ∈ 𝑋 𝑆 ⊆ 𝐴)
3	2	ancoms 255	. . . 4 ⊢ ((∀𝑥 ∈ 𝑋 𝑆 ⊆ 𝐴 ∧ ∃𝑥 𝑥 ∈ 𝑋) → ∃𝑥 ∈ 𝑋 𝑆 ⊆ 𝐴)
4		iinss 3708	. . . 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 2931	. . 3 ⊢ (∩ 𝑥 ∈ 𝑋 𝑆 ⊆ 𝐴 ↔ (∩ 𝑥 ∈ 𝑋 𝑆 ∩ 𝐴) = ∩ 𝑥 ∈ 𝑋 𝑆)
7	5, 6	sylib 127	. 2 ⊢ ((∀𝑥 ∈ 𝑋 𝑆 ⊆ 𝐴 ∧ ∃𝑥 𝑥 ∈ 𝑋) → (∩ 𝑥 ∈ 𝑋 𝑆 ∩ 𝐴) = ∩ 𝑥 ∈ 𝑋 𝑆)
8	1, 7	syl5eq 2084	1 ⊢ ((∀𝑥 ∈ 𝑋 𝑆 ⊆ 𝐴 ∧ ∃𝑥 𝑥 ∈ 𝑋) → (𝐴 ∩ ∩ 𝑥 ∈ 𝑋 𝑆) = ∩ 𝑥 ∈ 𝑋 𝑆)

Syntax hints:   → wi 4   ∧ wa 97   = wceq 1243  ∃wex 1381   ∈ wcel 1393  ∀wral 2306  ∃wrex 2307   ∩ cin 2916   ⊆ wss 2917  ∩ ciin 3658

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-iin 3660

This theorem is referenced by:  riinerm  6179