Theorem mressmrcd 17646

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 17642	. . . 4 ⊢ (𝜑 → (𝑁‘𝑇) ⊆ 𝑋)
5	1, 2, 3, 4	mrcssd 17643	. . 3 ⊢ (𝜑 → (𝑁‘𝑆) ⊆ (𝑁‘(𝑁‘𝑇)))
6		mressmrcd.4	. . . . 5 ⊢ (𝜑 → 𝑇 ⊆ 𝑆)
7	3, 4	sstrd 3976	. . . . 5 ⊢ (𝜑 → 𝑆 ⊆ 𝑋)
8	6, 7	sstrd 3976	. . . 4 ⊢ (𝜑 → 𝑇 ⊆ 𝑋)
9	1, 2, 8	mrcidmd 17645	. . 3 ⊢ (𝜑 → (𝑁‘(𝑁‘𝑇)) = (𝑁‘𝑇))
10	5, 9	sseqtrd 4002	. 2 ⊢ (𝜑 → (𝑁‘𝑆) ⊆ (𝑁‘𝑇))
11	1, 2, 6, 7	mrcssd 17643	. 2 ⊢ (𝜑 → (𝑁‘𝑇) ⊆ (𝑁‘𝑆))
12	10, 11	eqssd 3983	1 ⊢ (𝜑 → (𝑁‘𝑆) = (𝑁‘𝑇))

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2107   ⊆ wss 3933  ‘cfv 6542  Moorecmre 17601  mrClscmrc 17602

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-sep 5278  ax-nul 5288  ax-pow 5347  ax-pr 5414  ax-un 7738

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-rab 3421  df-v 3466  df-sbc 3773  df-csb 3882  df-dif 3936  df-un 3938  df-in 3940  df-ss 3950  df-nul 4316  df-if 4508  df-pw 4584  df-sn 4609  df-pr 4611  df-op 4615  df-uni 4890  df-int 4929  df-br 5126  df-opab 5188  df-mpt 5208  df-id 5560  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-iota 6495  df-fun 6544  df-fn 6545  df-f 6546  df-fv 6550  df-mre 17605  df-mrc 17606

This theorem is referenced by:  mrieqvlemd  17648  mrissmrcd  17659