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Theorem mrieqvlemd 17572
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 17581 and mrieqv2d 17582. 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 480 . . . 4 ((𝜑𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))) → 𝐴 ∈ (Moore‘𝑋))
3 mrieqvlemd.2 . . . 4 𝑁 = (mrCls‘𝐴)
4 undif1 4467 . . . . . 6 ((𝑆 ∖ {𝑌}) ∪ {𝑌}) = (𝑆 ∪ {𝑌})
5 mrieqvlemd.3 . . . . . . . . . 10 (𝜑𝑆𝑋)
65adantr 480 . . . . . . . . 9 ((𝜑𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))) → 𝑆𝑋)
76ssdifssd 4134 . . . . . . . 8 ((𝜑𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))) → (𝑆 ∖ {𝑌}) ⊆ 𝑋)
82, 3, 7mrcssidd 17568 . . . . . . 7 ((𝜑𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))) → (𝑆 ∖ {𝑌}) ⊆ (𝑁‘(𝑆 ∖ {𝑌})))
9 simpr 484 . . . . . . . 8 ((𝜑𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))) → 𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌})))
109snssd 4804 . . . . . . 7 ((𝜑𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))) → {𝑌} ⊆ (𝑁‘(𝑆 ∖ {𝑌})))
118, 10unssd 4178 . . . . . 6 ((𝜑𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))) → ((𝑆 ∖ {𝑌}) ∪ {𝑌}) ⊆ (𝑁‘(𝑆 ∖ {𝑌})))
124, 11eqsstrrid 4023 . . . . 5 ((𝜑𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))) → (𝑆 ∪ {𝑌}) ⊆ (𝑁‘(𝑆 ∖ {𝑌})))
1312unssad 4179 . . . 4 ((𝜑𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))) → 𝑆 ⊆ (𝑁‘(𝑆 ∖ {𝑌})))
14 difssd 4124 . . . 4 ((𝜑𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))) → (𝑆 ∖ {𝑌}) ⊆ 𝑆)
152, 3, 13, 14mressmrcd 17570 . . 3 ((𝜑𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))) → (𝑁𝑆) = (𝑁‘(𝑆 ∖ {𝑌})))
1615eqcomd 2730 . 2 ((𝜑𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))) → (𝑁‘(𝑆 ∖ {𝑌})) = (𝑁𝑆))
171, 3, 5mrcssidd 17568 . . . . 5 (𝜑𝑆 ⊆ (𝑁𝑆))
18 mrieqvlemd.4 . . . . 5 (𝜑𝑌𝑆)
1917, 18sseldd 3975 . . . 4 (𝜑𝑌 ∈ (𝑁𝑆))
2019adantr 480 . . 3 ((𝜑 ∧ (𝑁‘(𝑆 ∖ {𝑌})) = (𝑁𝑆)) → 𝑌 ∈ (𝑁𝑆))
21 simpr 484 . . 3 ((𝜑 ∧ (𝑁‘(𝑆 ∖ {𝑌})) = (𝑁𝑆)) → (𝑁‘(𝑆 ∖ {𝑌})) = (𝑁𝑆))
2220, 21eleqtrrd 2828 . 2 ((𝜑 ∧ (𝑁‘(𝑆 ∖ {𝑌})) = (𝑁𝑆)) → 𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌})))
2316, 22impbida 798 1 (𝜑 → (𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌})) ↔ (𝑁‘(𝑆 ∖ {𝑌})) = (𝑁𝑆)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1533  wcel 2098  cdif 3937  cun 3938  wss 3940  {csn 4620  cfv 6533  Moorecmre 17525  mrClscmrc 17526
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-int 4941  df-br 5139  df-opab 5201  df-mpt 5222  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-fv 6541  df-mre 17529  df-mrc 17530
This theorem is referenced by:  mrieqvd  17581  mrieqv2d  17582
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