Theorem mressmrcd 17594

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 17590	. . . 4 ⊢ (𝜑 → (𝑁‘𝑇) ⊆ 𝑋)
5	1, 2, 3, 4	mrcssd 17591	. . 3 ⊢ (𝜑 → (𝑁‘𝑆) ⊆ (𝑁‘(𝑁‘𝑇)))
6		mressmrcd.4	. . . . 5 ⊢ (𝜑 → 𝑇 ⊆ 𝑆)
7	3, 4	sstrd 3965	. . . . 5 ⊢ (𝜑 → 𝑆 ⊆ 𝑋)
8	6, 7	sstrd 3965	. . . 4 ⊢ (𝜑 → 𝑇 ⊆ 𝑋)
9	1, 2, 8	mrcidmd 17593	. . 3 ⊢ (𝜑 → (𝑁‘(𝑁‘𝑇)) = (𝑁‘𝑇))
10	5, 9	sseqtrd 3991	. 2 ⊢ (𝜑 → (𝑁‘𝑆) ⊆ (𝑁‘𝑇))
11	1, 2, 6, 7	mrcssd 17591	. 2 ⊢ (𝜑 → (𝑁‘𝑇) ⊆ (𝑁‘𝑆))
12	10, 11	eqssd 3972	1 ⊢ (𝜑 → (𝑁‘𝑆) = (𝑁‘𝑇))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109   ⊆ wss 3922  ‘cfv 6519  Moorecmre 17549  mrClscmrc 17550

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 2702  ax-sep 5259  ax-nul 5269  ax-pow 5328  ax-pr 5395  ax-un 7718

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 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2880  df-ne 2928  df-ral 3047  df-rex 3056  df-rab 3412  df-v 3457  df-sbc 3762  df-csb 3871  df-dif 3925  df-un 3927  df-in 3929  df-ss 3939  df-nul 4305  df-if 4497  df-pw 4573  df-sn 4598  df-pr 4600  df-op 4604  df-uni 4880  df-int 4919  df-br 5116  df-opab 5178  df-mpt 5197  df-id 5541  df-xp 5652  df-rel 5653  df-cnv 5654  df-co 5655  df-dm 5656  df-rn 5657  df-res 5658  df-ima 5659  df-iota 6472  df-fun 6521  df-fn 6522  df-f 6523  df-fv 6527  df-mre 17553  df-mrc 17554

This theorem is referenced by:  mrieqvlemd  17596  mrissmrcd  17607