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Theorem mrieqvlemd 16733
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 16742 and mrieqv2d 16743. 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 4344 . . . . . 6 ((𝑆 ∖ {𝑌}) ∪ {𝑌}) = (𝑆 ∪ {𝑌})
5 mrieqvlemd.3 . . . . . . . . . 10 (𝜑𝑆𝑋)
65adantr 481 . . . . . . . . 9 ((𝜑𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))) → 𝑆𝑋)
76ssdifssd 4046 . . . . . . . 8 ((𝜑𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))) → (𝑆 ∖ {𝑌}) ⊆ 𝑋)
82, 3, 7mrcssidd 16729 . . . . . . 7 ((𝜑𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))) → (𝑆 ∖ {𝑌}) ⊆ (𝑁‘(𝑆 ∖ {𝑌})))
9 simpr 485 . . . . . . . 8 ((𝜑𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))) → 𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌})))
109snssd 4655 . . . . . . 7 ((𝜑𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))) → {𝑌} ⊆ (𝑁‘(𝑆 ∖ {𝑌})))
118, 10unssd 4089 . . . . . 6 ((𝜑𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))) → ((𝑆 ∖ {𝑌}) ∪ {𝑌}) ⊆ (𝑁‘(𝑆 ∖ {𝑌})))
124, 11eqsstrrid 3943 . . . . 5 ((𝜑𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))) → (𝑆 ∪ {𝑌}) ⊆ (𝑁‘(𝑆 ∖ {𝑌})))
1312unssad 4090 . . . 4 ((𝜑𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))) → 𝑆 ⊆ (𝑁‘(𝑆 ∖ {𝑌})))
14 difssd 4036 . . . 4 ((𝜑𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))) → (𝑆 ∖ {𝑌}) ⊆ 𝑆)
152, 3, 13, 14mressmrcd 16731 . . 3 ((𝜑𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))) → (𝑁𝑆) = (𝑁‘(𝑆 ∖ {𝑌})))
1615eqcomd 2803 . 2 ((𝜑𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))) → (𝑁‘(𝑆 ∖ {𝑌})) = (𝑁𝑆))
171, 3, 5mrcssidd 16729 . . . . 5 (𝜑𝑆 ⊆ (𝑁𝑆))
18 mrieqvlemd.4 . . . . 5 (𝜑𝑌𝑆)
1917, 18sseldd 3896 . . . 4 (𝜑𝑌 ∈ (𝑁𝑆))
2019adantr 481 . . 3 ((𝜑 ∧ (𝑁‘(𝑆 ∖ {𝑌})) = (𝑁𝑆)) → 𝑌 ∈ (𝑁𝑆))
21 simpr 485 . . 3 ((𝜑 ∧ (𝑁‘(𝑆 ∖ {𝑌})) = (𝑁𝑆)) → (𝑁‘(𝑆 ∖ {𝑌})) = (𝑁𝑆))
2220, 21eleqtrrd 2888 . 2 ((𝜑 ∧ (𝑁‘(𝑆 ∖ {𝑌})) = (𝑁𝑆)) → 𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌})))
2316, 22impbida 797 1 (𝜑 → (𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌})) ↔ (𝑁‘(𝑆 ∖ {𝑌})) = (𝑁𝑆)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1525  wcel 2083  cdif 3862  cun 3863  wss 3865  {csn 4478  cfv 6232  Moorecmre 16686  mrClscmrc 16687
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-sep 5101  ax-nul 5108  ax-pow 5164  ax-pr 5228  ax-un 7326
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ne 2987  df-ral 3112  df-rex 3113  df-rab 3116  df-v 3442  df-sbc 3712  df-csb 3818  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-nul 4218  df-if 4388  df-pw 4461  df-sn 4479  df-pr 4481  df-op 4485  df-uni 4752  df-int 4789  df-br 4969  df-opab 5031  df-mpt 5048  df-id 5355  df-xp 5456  df-rel 5457  df-cnv 5458  df-co 5459  df-dm 5460  df-rn 5461  df-res 5462  df-ima 5463  df-iota 6196  df-fun 6234  df-fn 6235  df-f 6236  df-fv 6240  df-mre 16690  df-mrc 16691
This theorem is referenced by:  mrieqvd  16742  mrieqv2d  16743
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