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Theorem mressmrcd 17588

Description: In a Moore system, if a set is between another set and its closure, the two sets have the same closure. Deduction form. (Contributed by David Moews, 1-May-2017.)

**Hypotheses**
Ref	Expression
mressmrcd.1	⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋))
mressmrcd.2	⊢ 𝑁 = (mrCls‘𝐴)
mressmrcd.3	⊢ (𝜑 → 𝑆 ⊆ (𝑁‘𝑇))
mressmrcd.4	⊢ (𝜑 → 𝑇 ⊆ 𝑆)

**Assertion**
Ref	Expression
mressmrcd	⊢ (𝜑 → (𝑁‘𝑆) = (𝑁‘𝑇))

**Proof of Theorem mressmrcd**
Step	Hyp	Ref	Expression
1		mressmrcd.1	. . . 4 ⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋))
2		mressmrcd.2	. . . 4 ⊢ 𝑁 = (mrCls‘𝐴)
3		mressmrcd.3	. . . 4 ⊢ (𝜑 → 𝑆 ⊆ (𝑁‘𝑇))
4	1, 2	mrcssvd 17584	. . . 4 ⊢ (𝜑 → (𝑁‘𝑇) ⊆ 𝑋)
5	1, 2, 3, 4	mrcssd 17585	. . 3 ⊢ (𝜑 → (𝑁‘𝑆) ⊆ (𝑁‘(𝑁‘𝑇)))
6		mressmrcd.4	. . . . 5 ⊢ (𝜑 → 𝑇 ⊆ 𝑆)
7	3, 4	sstrd 3957	. . . . 5 ⊢ (𝜑 → 𝑆 ⊆ 𝑋)
8	6, 7	sstrd 3957	. . . 4 ⊢ (𝜑 → 𝑇 ⊆ 𝑋)
9	1, 2, 8	mrcidmd 17587	. . 3 ⊢ (𝜑 → (𝑁‘(𝑁‘𝑇)) = (𝑁‘𝑇))
10	5, 9	sseqtrd 3983	. 2 ⊢ (𝜑 → (𝑁‘𝑆) ⊆ (𝑁‘𝑇))
11	1, 2, 6, 7	mrcssd 17585	. 2 ⊢ (𝜑 → (𝑁‘𝑇) ⊆ (𝑁‘𝑆))
12	10, 11	eqssd 3964	1 ⊢ (𝜑 → (𝑁‘𝑆) = (𝑁‘𝑇))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2109 ⊆ wss 3914 ‘cfv 6511 Moorecmre 17543 mrClscmrc 17544

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-ral 3045 df-rex 3054 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4872 df-int 4911 df-br 5108 df-opab 5170 df-mpt 5189 df-id 5533 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-fv 6519 df-mre 17547 df-mrc 17548

This theorem is referenced by: mrieqvlemd 17590 mrissmrcd 17601

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