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Theorem mrieqvlemd 17578
Description: In a Moore system, if π‘Œ is a member of 𝑆, (𝑆 βˆ– {π‘Œ}) and 𝑆 have the same closure if and only if π‘Œ is in the closure of (𝑆 βˆ– {π‘Œ}). Used in the proof of mrieqvd 17587 and mrieqv2d 17588. Deduction form. (Contributed by David Moews, 1-May-2017.)
Hypotheses
Ref Expression
mrieqvlemd.1 (πœ‘ β†’ 𝐴 ∈ (Mooreβ€˜π‘‹))
mrieqvlemd.2 𝑁 = (mrClsβ€˜π΄)
mrieqvlemd.3 (πœ‘ β†’ 𝑆 βŠ† 𝑋)
mrieqvlemd.4 (πœ‘ β†’ π‘Œ ∈ 𝑆)
Assertion
Ref Expression
mrieqvlemd (πœ‘ β†’ (π‘Œ ∈ (π‘β€˜(𝑆 βˆ– {π‘Œ})) ↔ (π‘β€˜(𝑆 βˆ– {π‘Œ})) = (π‘β€˜π‘†)))
Proof of Theorem mrieqvlemd
StepHypRef Expression
1 mrieqvlemd.1 . . . . 5 (πœ‘ β†’ 𝐴 ∈ (Mooreβ€˜π‘‹))
21adantr 480 . . . 4 ((πœ‘ ∧ π‘Œ ∈ (π‘β€˜(𝑆 βˆ– {π‘Œ}))) β†’ 𝐴 ∈ (Mooreβ€˜π‘‹))
3 mrieqvlemd.2 . . . 4 𝑁 = (mrClsβ€˜π΄)
4 undif1 4475 . . . . . 6 ((𝑆 βˆ– {π‘Œ}) βˆͺ {π‘Œ}) = (𝑆 βˆͺ {π‘Œ})
5 mrieqvlemd.3 . . . . . . . . . 10 (πœ‘ β†’ 𝑆 βŠ† 𝑋)
65adantr 480 . . . . . . . . 9 ((πœ‘ ∧ π‘Œ ∈ (π‘β€˜(𝑆 βˆ– {π‘Œ}))) β†’ 𝑆 βŠ† 𝑋)
76ssdifssd 4142 . . . . . . . 8 ((πœ‘ ∧ π‘Œ ∈ (π‘β€˜(𝑆 βˆ– {π‘Œ}))) β†’ (𝑆 βˆ– {π‘Œ}) βŠ† 𝑋)
82, 3, 7mrcssidd 17574 . . . . . . 7 ((πœ‘ ∧ π‘Œ ∈ (π‘β€˜(𝑆 βˆ– {π‘Œ}))) β†’ (𝑆 βˆ– {π‘Œ}) βŠ† (π‘β€˜(𝑆 βˆ– {π‘Œ})))
9 simpr 484 . . . . . . . 8 ((πœ‘ ∧ π‘Œ ∈ (π‘β€˜(𝑆 βˆ– {π‘Œ}))) β†’ π‘Œ ∈ (π‘β€˜(𝑆 βˆ– {π‘Œ})))
109snssd 4812 . . . . . . 7 ((πœ‘ ∧ π‘Œ ∈ (π‘β€˜(𝑆 βˆ– {π‘Œ}))) β†’ {π‘Œ} βŠ† (π‘β€˜(𝑆 βˆ– {π‘Œ})))
118, 10unssd 4186 . . . . . 6 ((πœ‘ ∧ π‘Œ ∈ (π‘β€˜(𝑆 βˆ– {π‘Œ}))) β†’ ((𝑆 βˆ– {π‘Œ}) βˆͺ {π‘Œ}) βŠ† (π‘β€˜(𝑆 βˆ– {π‘Œ})))
124, 11eqsstrrid 4031 . . . . 5 ((πœ‘ ∧ π‘Œ ∈ (π‘β€˜(𝑆 βˆ– {π‘Œ}))) β†’ (𝑆 βˆͺ {π‘Œ}) βŠ† (π‘β€˜(𝑆 βˆ– {π‘Œ})))
1312unssad 4187 . . . 4 ((πœ‘ ∧ π‘Œ ∈ (π‘β€˜(𝑆 βˆ– {π‘Œ}))) β†’ 𝑆 βŠ† (π‘β€˜(𝑆 βˆ– {π‘Œ})))
14 difssd 4132 . . . 4 ((πœ‘ ∧ π‘Œ ∈ (π‘β€˜(𝑆 βˆ– {π‘Œ}))) β†’ (𝑆 βˆ– {π‘Œ}) βŠ† 𝑆)
152, 3, 13, 14mressmrcd 17576 . . 3 ((πœ‘ ∧ π‘Œ ∈ (π‘β€˜(𝑆 βˆ– {π‘Œ}))) β†’ (π‘β€˜π‘†) = (π‘β€˜(𝑆 βˆ– {π‘Œ})))
1615eqcomd 2737 . 2 ((πœ‘ ∧ π‘Œ ∈ (π‘β€˜(𝑆 βˆ– {π‘Œ}))) β†’ (π‘β€˜(𝑆 βˆ– {π‘Œ})) = (π‘β€˜π‘†))
171, 3, 5mrcssidd 17574 . . . . 5 (πœ‘ β†’ 𝑆 βŠ† (π‘β€˜π‘†))
18 mrieqvlemd.4 . . . . 5 (πœ‘ β†’ π‘Œ ∈ 𝑆)
1917, 18sseldd 3983 . . . 4 (πœ‘ β†’ π‘Œ ∈ (π‘β€˜π‘†))
2019adantr 480 . . 3 ((πœ‘ ∧ (π‘β€˜(𝑆 βˆ– {π‘Œ})) = (π‘β€˜π‘†)) β†’ π‘Œ ∈ (π‘β€˜π‘†))
21 simpr 484 . . 3 ((πœ‘ ∧ (π‘β€˜(𝑆 βˆ– {π‘Œ})) = (π‘β€˜π‘†)) β†’ (π‘β€˜(𝑆 βˆ– {π‘Œ})) = (π‘β€˜π‘†))
2220, 21eleqtrrd 2835 . 2 ((πœ‘ ∧ (π‘β€˜(𝑆 βˆ– {π‘Œ})) = (π‘β€˜π‘†)) β†’ π‘Œ ∈ (π‘β€˜(𝑆 βˆ– {π‘Œ})))
2316, 22impbida 798 1 (πœ‘ β†’ (π‘Œ ∈ (π‘β€˜(𝑆 βˆ– {π‘Œ})) ↔ (π‘β€˜(𝑆 βˆ– {π‘Œ})) = (π‘β€˜π‘†)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   = wceq 1540   ∈ wcel 2105   βˆ– cdif 3945   βˆͺ cun 3946   βŠ† wss 3948  {csn 4628  β€˜cfv 6543  Moorecmre 17531  mrClscmrc 17532
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-mre 17535  df-mrc 17536
This theorem is referenced by:  mrieqvd  17587  mrieqv2d  17588
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