Theorem mressmrcd 17659

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 17655	. . . 4 ⊢ (𝜑 → (𝑁‘𝑇) ⊆ 𝑋)
5	1, 2, 3, 4	mrcssd 17656	. . 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 17658	. . 3 ⊢ (𝜑 → (𝑁‘(𝑁‘𝑇)) = (𝑁‘𝑇))
10	5, 9	sseqtrd 3972	. 2 ⊢ (𝜑 → (𝑁‘𝑆) ⊆ (𝑁‘𝑇))
11	1, 2, 6, 7	mrcssd 17656	. 2 ⊢ (𝜑 → (𝑁‘𝑇) ⊆ (𝑁‘𝑆))
12	10, 11	eqssd 3953	1 ⊢ (𝜑 → (𝑁‘𝑆) = (𝑁‘𝑇))

Syntax hints:   → wi 4   = wceq 1560   ∈ wcel 2142   ⊆ wss 3904  ‘cfv 6521  Moorecmre 17610  mrClscmrc 17611

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5246  ax-nul 5256  ax-pow 5322  ax-pr 5390  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4906  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-fv 6529  df-mre 17614  df-mrc 17615

This theorem is referenced by:  mrieqvlemd  17661  mrissmrcd  17672