Theorem submre 17231

Description: The subcollection of a closed set system below a given closed set is itself a closed set system. (Contributed by Stefan O'Rear, 9-Mar-2015.)

Assertion

Ref	Expression
submre	⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐴 ∈ 𝐶) → (𝐶 ∩ 𝒫 𝐴) ∈ (Moore‘𝐴))

Proof of Theorem submre

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		inss2 4160	. . 3 ⊢ (𝐶 ∩ 𝒫 𝐴) ⊆ 𝒫 𝐴
2	1	a1i 11	. 2 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐴 ∈ 𝐶) → (𝐶 ∩ 𝒫 𝐴) ⊆ 𝒫 𝐴)
3		simpr 484	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐴 ∈ 𝐶) → 𝐴 ∈ 𝐶)
4		pwidg 4552	. . . 4 ⊢ (𝐴 ∈ 𝐶 → 𝐴 ∈ 𝒫 𝐴)
5	4	adantl 481	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐴 ∈ 𝐶) → 𝐴 ∈ 𝒫 𝐴)
6	3, 5	elind 4124	. 2 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐴 ∈ 𝐶) → 𝐴 ∈ (𝐶 ∩ 𝒫 𝐴))
7		simp1l 1195	. . . 4 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝐴 ∈ 𝐶) ∧ 𝑥 ⊆ (𝐶 ∩ 𝒫 𝐴) ∧ 𝑥 ≠ ∅) → 𝐶 ∈ (Moore‘𝑋))
8		inss1 4159	. . . . . 6 ⊢ (𝐶 ∩ 𝒫 𝐴) ⊆ 𝐶
9		sstr 3925	. . . . . 6 ⊢ ((𝑥 ⊆ (𝐶 ∩ 𝒫 𝐴) ∧ (𝐶 ∩ 𝒫 𝐴) ⊆ 𝐶) → 𝑥 ⊆ 𝐶)
10	8, 9	mpan2 687	. . . . 5 ⊢ (𝑥 ⊆ (𝐶 ∩ 𝒫 𝐴) → 𝑥 ⊆ 𝐶)
11	10	3ad2ant2 1132	. . . 4 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝐴 ∈ 𝐶) ∧ 𝑥 ⊆ (𝐶 ∩ 𝒫 𝐴) ∧ 𝑥 ≠ ∅) → 𝑥 ⊆ 𝐶)
12		simp3 1136	. . . 4 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝐴 ∈ 𝐶) ∧ 𝑥 ⊆ (𝐶 ∩ 𝒫 𝐴) ∧ 𝑥 ≠ ∅) → 𝑥 ≠ ∅)
13		mreintcl 17221	. . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥 ⊆ 𝐶 ∧ 𝑥 ≠ ∅) → ∩ 𝑥 ∈ 𝐶)
14	7, 11, 12, 13	syl3anc 1369	. . 3 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝐴 ∈ 𝐶) ∧ 𝑥 ⊆ (𝐶 ∩ 𝒫 𝐴) ∧ 𝑥 ≠ ∅) → ∩ 𝑥 ∈ 𝐶)
15		sstr 3925	. . . . . . . 8 ⊢ ((𝑥 ⊆ (𝐶 ∩ 𝒫 𝐴) ∧ (𝐶 ∩ 𝒫 𝐴) ⊆ 𝒫 𝐴) → 𝑥 ⊆ 𝒫 𝐴)
16	1, 15	mpan2 687	. . . . . . 7 ⊢ (𝑥 ⊆ (𝐶 ∩ 𝒫 𝐴) → 𝑥 ⊆ 𝒫 𝐴)
17	16	3ad2ant2 1132	. . . . . 6 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝐴 ∈ 𝐶) ∧ 𝑥 ⊆ (𝐶 ∩ 𝒫 𝐴) ∧ 𝑥 ≠ ∅) → 𝑥 ⊆ 𝒫 𝐴)
18		intssuni2 4901	. . . . . 6 ⊢ ((𝑥 ⊆ 𝒫 𝐴 ∧ 𝑥 ≠ ∅) → ∩ 𝑥 ⊆ ∪ 𝒫 𝐴)
19	17, 12, 18	syl2anc 583	. . . . 5 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝐴 ∈ 𝐶) ∧ 𝑥 ⊆ (𝐶 ∩ 𝒫 𝐴) ∧ 𝑥 ≠ ∅) → ∩ 𝑥 ⊆ ∪ 𝒫 𝐴)
20		unipw 5360	. . . . 5 ⊢ ∪ 𝒫 𝐴 = 𝐴
21	19, 20	sseqtrdi 3967	. . . 4 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝐴 ∈ 𝐶) ∧ 𝑥 ⊆ (𝐶 ∩ 𝒫 𝐴) ∧ 𝑥 ≠ ∅) → ∩ 𝑥 ⊆ 𝐴)
22		elpw2g 5263	. . . . . 6 ⊢ (𝐴 ∈ 𝐶 → (∩ 𝑥 ∈ 𝒫 𝐴 ↔ ∩ 𝑥 ⊆ 𝐴))
23	22	adantl 481	. . . . 5 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐴 ∈ 𝐶) → (∩ 𝑥 ∈ 𝒫 𝐴 ↔ ∩ 𝑥 ⊆ 𝐴))
24	23	3ad2ant1 1131	. . . 4 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝐴 ∈ 𝐶) ∧ 𝑥 ⊆ (𝐶 ∩ 𝒫 𝐴) ∧ 𝑥 ≠ ∅) → (∩ 𝑥 ∈ 𝒫 𝐴 ↔ ∩ 𝑥 ⊆ 𝐴))
25	21, 24	mpbird 256	. . 3 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝐴 ∈ 𝐶) ∧ 𝑥 ⊆ (𝐶 ∩ 𝒫 𝐴) ∧ 𝑥 ≠ ∅) → ∩ 𝑥 ∈ 𝒫 𝐴)
26	14, 25	elind 4124	. 2 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝐴 ∈ 𝐶) ∧ 𝑥 ⊆ (𝐶 ∩ 𝒫 𝐴) ∧ 𝑥 ≠ ∅) → ∩ 𝑥 ∈ (𝐶 ∩ 𝒫 𝐴))
27	2, 6, 26	ismred 17228	1 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐴 ∈ 𝐶) → (𝐶 ∩ 𝒫 𝐴) ∈ (Moore‘𝐴))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   ∈ wcel 2108   ≠ wne 2942   ∩ cin 3882   ⊆ wss 3883  ∅c0 4253  𝒫 cpw 4530  ∪ cuni 4836  ∩ cint 4876  ‘cfv 6418  Moorecmre 17208

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-int 4877  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-iota 6376  df-fun 6420  df-fv 6426  df-mre 17212

This theorem is referenced by:  submrc  17254