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Theorem mrieqvlemd 16497
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 16506 and mrieqv2d 16507. 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 466 . . . 4 ((𝜑𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))) → 𝐴 ∈ (Moore‘𝑋))
3 mrieqvlemd.2 . . . 4 𝑁 = (mrCls‘𝐴)
4 undif1 4185 . . . . . 6 ((𝑆 ∖ {𝑌}) ∪ {𝑌}) = (𝑆 ∪ {𝑌})
5 mrieqvlemd.3 . . . . . . . . . 10 (𝜑𝑆𝑋)
65adantr 466 . . . . . . . . 9 ((𝜑𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))) → 𝑆𝑋)
76ssdifssd 3899 . . . . . . . 8 ((𝜑𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))) → (𝑆 ∖ {𝑌}) ⊆ 𝑋)
82, 3, 7mrcssidd 16493 . . . . . . 7 ((𝜑𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))) → (𝑆 ∖ {𝑌}) ⊆ (𝑁‘(𝑆 ∖ {𝑌})))
9 simpr 471 . . . . . . . 8 ((𝜑𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))) → 𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌})))
109snssd 4475 . . . . . . 7 ((𝜑𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))) → {𝑌} ⊆ (𝑁‘(𝑆 ∖ {𝑌})))
118, 10unssd 3940 . . . . . 6 ((𝜑𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))) → ((𝑆 ∖ {𝑌}) ∪ {𝑌}) ⊆ (𝑁‘(𝑆 ∖ {𝑌})))
124, 11syl5eqssr 3799 . . . . 5 ((𝜑𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))) → (𝑆 ∪ {𝑌}) ⊆ (𝑁‘(𝑆 ∖ {𝑌})))
1312unssad 3941 . . . 4 ((𝜑𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))) → 𝑆 ⊆ (𝑁‘(𝑆 ∖ {𝑌})))
14 difssd 3889 . . . 4 ((𝜑𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))) → (𝑆 ∖ {𝑌}) ⊆ 𝑆)
152, 3, 13, 14mressmrcd 16495 . . 3 ((𝜑𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))) → (𝑁𝑆) = (𝑁‘(𝑆 ∖ {𝑌})))
1615eqcomd 2777 . 2 ((𝜑𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))) → (𝑁‘(𝑆 ∖ {𝑌})) = (𝑁𝑆))
171, 3, 5mrcssidd 16493 . . . . 5 (𝜑𝑆 ⊆ (𝑁𝑆))
18 mrieqvlemd.4 . . . . 5 (𝜑𝑌𝑆)
1917, 18sseldd 3753 . . . 4 (𝜑𝑌 ∈ (𝑁𝑆))
2019adantr 466 . . 3 ((𝜑 ∧ (𝑁‘(𝑆 ∖ {𝑌})) = (𝑁𝑆)) → 𝑌 ∈ (𝑁𝑆))
21 simpr 471 . . 3 ((𝜑 ∧ (𝑁‘(𝑆 ∖ {𝑌})) = (𝑁𝑆)) → (𝑁‘(𝑆 ∖ {𝑌})) = (𝑁𝑆))
2220, 21eleqtrrd 2853 . 2 ((𝜑 ∧ (𝑁‘(𝑆 ∖ {𝑌})) = (𝑁𝑆)) → 𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌})))
2316, 22impbida 802 1 (𝜑 → (𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌})) ↔ (𝑁‘(𝑆 ∖ {𝑌})) = (𝑁𝑆)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1631  wcel 2145  cdif 3720  cun 3721  wss 3723  {csn 4316  cfv 6031  Moorecmre 16450  mrClscmrc 16451
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-int 4612  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-fv 6039  df-mre 16454  df-mrc 16455
This theorem is referenced by:  mrieqvd  16506  mrieqv2d  16507
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