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Theorem mrieqvlemd 16229
Description: In a Moore system, if 𝑌 is a member of 𝑆, (𝑆 ∖ {𝑌}) and 𝑆 have the same closure if and only if 𝑌 is in the closure of (𝑆 ∖ {𝑌}). Used in the proof of mrieqvd 16238 and mrieqv2d 16239. Deduction form. (Contributed by David Moews, 1-May-2017.)
Hypotheses
Ref Expression
mrieqvlemd.1 (𝜑𝐴 ∈ (Moore‘𝑋))
mrieqvlemd.2 𝑁 = (mrCls‘𝐴)
mrieqvlemd.3 (𝜑𝑆𝑋)
mrieqvlemd.4 (𝜑𝑌𝑆)
Assertion
Ref Expression
mrieqvlemd (𝜑 → (𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌})) ↔ (𝑁‘(𝑆 ∖ {𝑌})) = (𝑁𝑆)))

Proof of Theorem mrieqvlemd
StepHypRef Expression
1 mrieqvlemd.1 . . . . 5 (𝜑𝐴 ∈ (Moore‘𝑋))
21adantr 481 . . . 4 ((𝜑𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))) → 𝐴 ∈ (Moore‘𝑋))
3 mrieqvlemd.2 . . . 4 𝑁 = (mrCls‘𝐴)
4 undif1 4021 . . . . . 6 ((𝑆 ∖ {𝑌}) ∪ {𝑌}) = (𝑆 ∪ {𝑌})
5 mrieqvlemd.3 . . . . . . . . . 10 (𝜑𝑆𝑋)
65adantr 481 . . . . . . . . 9 ((𝜑𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))) → 𝑆𝑋)
76ssdifssd 3732 . . . . . . . 8 ((𝜑𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))) → (𝑆 ∖ {𝑌}) ⊆ 𝑋)
82, 3, 7mrcssidd 16225 . . . . . . 7 ((𝜑𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))) → (𝑆 ∖ {𝑌}) ⊆ (𝑁‘(𝑆 ∖ {𝑌})))
9 simpr 477 . . . . . . . 8 ((𝜑𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))) → 𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌})))
109snssd 4316 . . . . . . 7 ((𝜑𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))) → {𝑌} ⊆ (𝑁‘(𝑆 ∖ {𝑌})))
118, 10unssd 3773 . . . . . 6 ((𝜑𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))) → ((𝑆 ∖ {𝑌}) ∪ {𝑌}) ⊆ (𝑁‘(𝑆 ∖ {𝑌})))
124, 11syl5eqssr 3635 . . . . 5 ((𝜑𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))) → (𝑆 ∪ {𝑌}) ⊆ (𝑁‘(𝑆 ∖ {𝑌})))
1312unssad 3774 . . . 4 ((𝜑𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))) → 𝑆 ⊆ (𝑁‘(𝑆 ∖ {𝑌})))
14 difssd 3722 . . . 4 ((𝜑𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))) → (𝑆 ∖ {𝑌}) ⊆ 𝑆)
152, 3, 13, 14mressmrcd 16227 . . 3 ((𝜑𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))) → (𝑁𝑆) = (𝑁‘(𝑆 ∖ {𝑌})))
1615eqcomd 2627 . 2 ((𝜑𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))) → (𝑁‘(𝑆 ∖ {𝑌})) = (𝑁𝑆))
171, 3, 5mrcssidd 16225 . . . . 5 (𝜑𝑆 ⊆ (𝑁𝑆))
18 mrieqvlemd.4 . . . . 5 (𝜑𝑌𝑆)
1917, 18sseldd 3589 . . . 4 (𝜑𝑌 ∈ (𝑁𝑆))
2019adantr 481 . . 3 ((𝜑 ∧ (𝑁‘(𝑆 ∖ {𝑌})) = (𝑁𝑆)) → 𝑌 ∈ (𝑁𝑆))
21 simpr 477 . . 3 ((𝜑 ∧ (𝑁‘(𝑆 ∖ {𝑌})) = (𝑁𝑆)) → (𝑁‘(𝑆 ∖ {𝑌})) = (𝑁𝑆))
2220, 21eleqtrrd 2701 . 2 ((𝜑 ∧ (𝑁‘(𝑆 ∖ {𝑌})) = (𝑁𝑆)) → 𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌})))
2316, 22impbida 876 1 (𝜑 → (𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌})) ↔ (𝑁‘(𝑆 ∖ {𝑌})) = (𝑁𝑆)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  cdif 3557  cun 3558  wss 3560  {csn 4155  cfv 5857  Moorecmre 16182  mrClscmrc 16183
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-int 4448  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-fv 5865  df-mre 16186  df-mrc 16187
This theorem is referenced by:  mrieqvd  16238  mrieqv2d  16239
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