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Theorem mressmrcd 16874
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 (𝜑𝐴 ∈ (Moore‘𝑋))
mressmrcd.2 𝑁 = (mrCls‘𝐴)
mressmrcd.3 (𝜑𝑆 ⊆ (𝑁𝑇))
mressmrcd.4 (𝜑𝑇𝑆)
Assertion
Ref Expression
mressmrcd (𝜑 → (𝑁𝑆) = (𝑁𝑇))

Proof of Theorem mressmrcd
StepHypRef Expression
1 mressmrcd.1 . . . 4 (𝜑𝐴 ∈ (Moore‘𝑋))
2 mressmrcd.2 . . . 4 𝑁 = (mrCls‘𝐴)
3 mressmrcd.3 . . . 4 (𝜑𝑆 ⊆ (𝑁𝑇))
41, 2mrcssvd 16870 . . . 4 (𝜑 → (𝑁𝑇) ⊆ 𝑋)
51, 2, 3, 4mrcssd 16871 . . 3 (𝜑 → (𝑁𝑆) ⊆ (𝑁‘(𝑁𝑇)))
6 mressmrcd.4 . . . . 5 (𝜑𝑇𝑆)
73, 4sstrd 3953 . . . . 5 (𝜑𝑆𝑋)
86, 7sstrd 3953 . . . 4 (𝜑𝑇𝑋)
91, 2, 8mrcidmd 16873 . . 3 (𝜑 → (𝑁‘(𝑁𝑇)) = (𝑁𝑇))
105, 9sseqtrd 3983 . 2 (𝜑 → (𝑁𝑆) ⊆ (𝑁𝑇))
111, 2, 6, 7mrcssd 16871 . 2 (𝜑 → (𝑁𝑇) ⊆ (𝑁𝑆))
1210, 11eqssd 3960 1 (𝜑 → (𝑁𝑆) = (𝑁𝑇))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2114  wss 3912  cfv 6329  Moorecmre 16829  mrClscmrc 16830
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-sep 5177  ax-nul 5184  ax-pow 5240  ax-pr 5304  ax-un 7437
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-ral 3130  df-rex 3131  df-rab 3134  df-v 3475  df-sbc 3752  df-csb 3860  df-dif 3915  df-un 3917  df-in 3919  df-ss 3928  df-nul 4268  df-if 4442  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4813  df-int 4851  df-br 5041  df-opab 5103  df-mpt 5121  df-id 5434  df-xp 5535  df-rel 5536  df-cnv 5537  df-co 5538  df-dm 5539  df-rn 5540  df-res 5541  df-ima 5542  df-iota 6288  df-fun 6331  df-fn 6332  df-f 6333  df-fv 6337  df-mre 16833  df-mrc 16834
This theorem is referenced by:  mrieqvlemd  16876  mrissmrcd  16887
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