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Theorem mrieqvlemd 16112
 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 16121 and mrieqv2d 16122. 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 3995 . . . . . 6 ((𝑆 ∖ {𝑌}) ∪ {𝑌}) = (𝑆 ∪ {𝑌})
5 mrieqvlemd.3 . . . . . . . . . 10 (𝜑𝑆𝑋)
65adantr 480 . . . . . . . . 9 ((𝜑𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))) → 𝑆𝑋)
76ssdifssd 3710 . . . . . . . 8 ((𝜑𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))) → (𝑆 ∖ {𝑌}) ⊆ 𝑋)
82, 3, 7mrcssidd 16108 . . . . . . 7 ((𝜑𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))) → (𝑆 ∖ {𝑌}) ⊆ (𝑁‘(𝑆 ∖ {𝑌})))
9 simpr 476 . . . . . . . 8 ((𝜑𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))) → 𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌})))
109snssd 4281 . . . . . . 7 ((𝜑𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))) → {𝑌} ⊆ (𝑁‘(𝑆 ∖ {𝑌})))
118, 10unssd 3751 . . . . . 6 ((𝜑𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))) → ((𝑆 ∖ {𝑌}) ∪ {𝑌}) ⊆ (𝑁‘(𝑆 ∖ {𝑌})))
124, 11syl5eqssr 3613 . . . . 5 ((𝜑𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))) → (𝑆 ∪ {𝑌}) ⊆ (𝑁‘(𝑆 ∖ {𝑌})))
1312unssad 3752 . . . 4 ((𝜑𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))) → 𝑆 ⊆ (𝑁‘(𝑆 ∖ {𝑌})))
14 difssd 3700 . . . 4 ((𝜑𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))) → (𝑆 ∖ {𝑌}) ⊆ 𝑆)
152, 3, 13, 14mressmrcd 16110 . . 3 ((𝜑𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))) → (𝑁𝑆) = (𝑁‘(𝑆 ∖ {𝑌})))
1615eqcomd 2616 . 2 ((𝜑𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))) → (𝑁‘(𝑆 ∖ {𝑌})) = (𝑁𝑆))
171, 3, 5mrcssidd 16108 . . . . 5 (𝜑𝑆 ⊆ (𝑁𝑆))
18 mrieqvlemd.4 . . . . 5 (𝜑𝑌𝑆)
1917, 18sseldd 3569 . . . 4 (𝜑𝑌 ∈ (𝑁𝑆))
2019adantr 480 . . 3 ((𝜑 ∧ (𝑁‘(𝑆 ∖ {𝑌})) = (𝑁𝑆)) → 𝑌 ∈ (𝑁𝑆))
21 simpr 476 . . 3 ((𝜑 ∧ (𝑁‘(𝑆 ∖ {𝑌})) = (𝑁𝑆)) → (𝑁‘(𝑆 ∖ {𝑌})) = (𝑁𝑆))
2220, 21eleqtrrd 2691 . 2 ((𝜑 ∧ (𝑁‘(𝑆 ∖ {𝑌})) = (𝑁𝑆)) → 𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌})))
2316, 22impbida 873 1 (𝜑 → (𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌})) ↔ (𝑁‘(𝑆 ∖ {𝑌})) = (𝑁𝑆)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ∖ cdif 3537   ∪ cun 3538   ⊆ wss 3540  {csn 4125  ‘cfv 5804  Moorecmre 16065  mrClscmrc 16066 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-int 4411  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-mre 16069  df-mrc 16070 This theorem is referenced by:  mrieqvd  16121  mrieqv2d  16122
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