Theorem submrc 17536

Description: In a closure system which is cut off above some level, closures below that level act as normal. (Contributed by Stefan O'Rear, 9-Mar-2015.)

Hypotheses

Ref	Expression
submrc.f	⊢ 𝐹 = (mrCls‘𝐶)
submrc.g	⊢ 𝐺 = (mrCls‘(𝐶 ∩ 𝒫 𝐷))

Assertion

Ref	Expression
submrc	⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷 ∈ 𝐶 ∧ 𝑈 ⊆ 𝐷) → (𝐺‘𝑈) = (𝐹‘𝑈))

Proof of Theorem submrc

Step	Hyp	Ref	Expression
1		submre 17509	. . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷 ∈ 𝐶) → (𝐶 ∩ 𝒫 𝐷) ∈ (Moore‘𝐷))
2	1	3adant3 1132	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷 ∈ 𝐶 ∧ 𝑈 ⊆ 𝐷) → (𝐶 ∩ 𝒫 𝐷) ∈ (Moore‘𝐷))
3		simp1 1136	. . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷 ∈ 𝐶 ∧ 𝑈 ⊆ 𝐷) → 𝐶 ∈ (Moore‘𝑋))
4		submrc.f	. . . 4 ⊢ 𝐹 = (mrCls‘𝐶)
5		simp3 1138	. . . . 5 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷 ∈ 𝐶 ∧ 𝑈 ⊆ 𝐷) → 𝑈 ⊆ 𝐷)
6		mress 17497	. . . . . 6 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷 ∈ 𝐶) → 𝐷 ⊆ 𝑋)
7	6	3adant3 1132	. . . . 5 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷 ∈ 𝐶 ∧ 𝑈 ⊆ 𝐷) → 𝐷 ⊆ 𝑋)
8	5, 7	sstrd 3941	. . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷 ∈ 𝐶 ∧ 𝑈 ⊆ 𝐷) → 𝑈 ⊆ 𝑋)
9	3, 4, 8	mrcssidd 17533	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷 ∈ 𝐶 ∧ 𝑈 ⊆ 𝐷) → 𝑈 ⊆ (𝐹‘𝑈))
10	4	mrccl 17519	. . . . 5 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋) → (𝐹‘𝑈) ∈ 𝐶)
11	3, 8, 10	syl2anc 584	. . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷 ∈ 𝐶 ∧ 𝑈 ⊆ 𝐷) → (𝐹‘𝑈) ∈ 𝐶)
12	4	mrcsscl 17528	. . . . . 6 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐷 ∧ 𝐷 ∈ 𝐶) → (𝐹‘𝑈) ⊆ 𝐷)
13	12	3com23 1126	. . . . 5 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷 ∈ 𝐶 ∧ 𝑈 ⊆ 𝐷) → (𝐹‘𝑈) ⊆ 𝐷)
14		fvex 6841	. . . . . 6 ⊢ (𝐹‘𝑈) ∈ V
15	14	elpw 4553	. . . . 5 ⊢ ((𝐹‘𝑈) ∈ 𝒫 𝐷 ↔ (𝐹‘𝑈) ⊆ 𝐷)
16	13, 15	sylibr 234	. . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷 ∈ 𝐶 ∧ 𝑈 ⊆ 𝐷) → (𝐹‘𝑈) ∈ 𝒫 𝐷)
17	11, 16	elind 4149	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷 ∈ 𝐶 ∧ 𝑈 ⊆ 𝐷) → (𝐹‘𝑈) ∈ (𝐶 ∩ 𝒫 𝐷))
18		submrc.g	. . . 4 ⊢ 𝐺 = (mrCls‘(𝐶 ∩ 𝒫 𝐷))
19	18	mrcsscl 17528	. . 3 ⊢ (((𝐶 ∩ 𝒫 𝐷) ∈ (Moore‘𝐷) ∧ 𝑈 ⊆ (𝐹‘𝑈) ∧ (𝐹‘𝑈) ∈ (𝐶 ∩ 𝒫 𝐷)) → (𝐺‘𝑈) ⊆ (𝐹‘𝑈))
20	2, 9, 17, 19	syl3anc 1373	. 2 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷 ∈ 𝐶 ∧ 𝑈 ⊆ 𝐷) → (𝐺‘𝑈) ⊆ (𝐹‘𝑈))
21	2, 18, 5	mrcssidd 17533	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷 ∈ 𝐶 ∧ 𝑈 ⊆ 𝐷) → 𝑈 ⊆ (𝐺‘𝑈))
22	18	mrccl 17519	. . . . 5 ⊢ (((𝐶 ∩ 𝒫 𝐷) ∈ (Moore‘𝐷) ∧ 𝑈 ⊆ 𝐷) → (𝐺‘𝑈) ∈ (𝐶 ∩ 𝒫 𝐷))
23	2, 5, 22	syl2anc 584	. . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷 ∈ 𝐶 ∧ 𝑈 ⊆ 𝐷) → (𝐺‘𝑈) ∈ (𝐶 ∩ 𝒫 𝐷))
24	23	elin1d 4153	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷 ∈ 𝐶 ∧ 𝑈 ⊆ 𝐷) → (𝐺‘𝑈) ∈ 𝐶)
25	4	mrcsscl 17528	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ (𝐺‘𝑈) ∧ (𝐺‘𝑈) ∈ 𝐶) → (𝐹‘𝑈) ⊆ (𝐺‘𝑈))
26	3, 21, 24, 25	syl3anc 1373	. 2 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷 ∈ 𝐶 ∧ 𝑈 ⊆ 𝐷) → (𝐹‘𝑈) ⊆ (𝐺‘𝑈))
27	20, 26	eqssd 3948	1 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷 ∈ 𝐶 ∧ 𝑈 ⊆ 𝐷) → (𝐺‘𝑈) = (𝐹‘𝑈))

Syntax hints:   → wi 4   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113   ∩ cin 3897   ⊆ wss 3898  𝒫 cpw 4549  ‘cfv 6486  Moorecmre 17486  mrClscmrc 17487

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-int 4898  df-br 5094  df-opab 5156  df-mpt 5175  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-fv 6494  df-mre 17490  df-mrc 17491

This theorem is referenced by:  evlseu  22019