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Theorem mressmrcd 17689

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 17685	. . . 4 ⊢ (𝜑 → (𝑁‘𝑇) ⊆ 𝑋)
5	1, 2, 3, 4	mrcssd 17686	. . 3 ⊢ (𝜑 → (𝑁‘𝑆) ⊆ (𝑁‘(𝑁‘𝑇)))
6		mressmrcd.4	. . . . 5 ⊢ (𝜑 → 𝑇 ⊆ 𝑆)
7	3, 4	sstrd 3946	. . . . 5 ⊢ (𝜑 → 𝑆 ⊆ 𝑋)
8	6, 7	sstrd 3946	. . . 4 ⊢ (𝜑 → 𝑇 ⊆ 𝑋)
9	1, 2, 8	mrcidmd 17688	. . 3 ⊢ (𝜑 → (𝑁‘(𝑁‘𝑇)) = (𝑁‘𝑇))
10	5, 9	sseqtrd 3972	. 2 ⊢ (𝜑 → (𝑁‘𝑆) ⊆ (𝑁‘𝑇))
11	1, 2, 6, 7	mrcssd 17686	. 2 ⊢ (𝜑 → (𝑁‘𝑇) ⊆ (𝑁‘𝑆))
12	10, 11	eqssd 3953	1 ⊢ (𝜑 → (𝑁‘𝑆) = (𝑁‘𝑇))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 = wceq 1569 ∈ wcel 2142 ⊆ wss 3904 ‘cfv 6536 Moorecmre 17640 mrClscmrc 17641

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1824 ax-4 1838 ax-5 1939 ax-6 1996 ax-7 2037 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-sep 5256 ax-nul 5268 ax-pow 5335 ax-pr 5403 ax-un 7734

This proof depends on definitions: df-bi 210 df-an 401 df-or 861 df-3an 1104 df-tru 1572 df-fal 1582 df-ex 1809 df-nf 1813 df-sb 2096 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-ral 3079 df-rex 3089 df-rab 3416 df-v 3456 df-sbc 3744 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-nul 4286 df-if 4487 df-pw 4563 df-sn 4589 df-pr 4591 df-op 4595 df-uni 4872 df-int 4912 df-br 5109 df-opab 5173 df-mpt 5192 df-id 5555 df-xp 5666 df-rel 5667 df-cnv 5668 df-co 5669 df-dm 5670 df-rn 5671 df-res 5672 df-ima 5673 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-fv 6544 df-mre 17644 df-mrc 17645

This theorem is used by: mrieqvlemd 17691 mrissmrcd 17702

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