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Theorem mreexd 16074
Description: In a Moore system, the closure operator is said to have the exchange property if, for all elements 𝑦 and 𝑧 of the base set and subsets 𝑆 of the base set such that 𝑧 is in the closure of (𝑆 ∪ {𝑦}) but not in the closure of 𝑆, 𝑦 is in the closure of (𝑆 ∪ {𝑧}) (Definition 3.1.9 in [FaureFrolicher] p. 57 to 58.) This theorem allows us to construct substitution instances of this definition. (Contributed by David Moews, 1-May-2017.)
Hypotheses
Ref Expression
mreexd.1 (𝜑𝑋𝑉)
mreexd.2 (𝜑 → ∀𝑠 ∈ 𝒫 𝑋𝑦𝑋𝑧 ∈ ((𝑁‘(𝑠 ∪ {𝑦})) ∖ (𝑁𝑠))𝑦 ∈ (𝑁‘(𝑠 ∪ {𝑧})))
mreexd.3 (𝜑𝑆𝑋)
mreexd.4 (𝜑𝑌𝑋)
mreexd.5 (𝜑𝑍 ∈ (𝑁‘(𝑆 ∪ {𝑌})))
mreexd.6 (𝜑 → ¬ 𝑍 ∈ (𝑁𝑆))
Assertion
Ref Expression
mreexd (𝜑𝑌 ∈ (𝑁‘(𝑆 ∪ {𝑍})))
Distinct variable groups:   𝑋,𝑠,𝑦   𝑆,𝑠,𝑧,𝑦   𝜑,𝑠,𝑦,𝑧   𝑌,𝑠,𝑦,𝑧   𝑍,𝑠,𝑦,𝑧   𝑁,𝑠,𝑦,𝑧
Allowed substitution hints:   𝑉(𝑦,𝑧,𝑠)   𝑋(𝑧)

Proof of Theorem mreexd
StepHypRef Expression
1 mreexd.2 . 2 (𝜑 → ∀𝑠 ∈ 𝒫 𝑋𝑦𝑋𝑧 ∈ ((𝑁‘(𝑠 ∪ {𝑦})) ∖ (𝑁𝑠))𝑦 ∈ (𝑁‘(𝑠 ∪ {𝑧})))
2 mreexd.3 . . . 4 (𝜑𝑆𝑋)
3 mreexd.1 . . . . 5 (𝜑𝑋𝑉)
4 elpw2g 4749 . . . . 5 (𝑋𝑉 → (𝑆 ∈ 𝒫 𝑋𝑆𝑋))
53, 4syl 17 . . . 4 (𝜑 → (𝑆 ∈ 𝒫 𝑋𝑆𝑋))
62, 5mpbird 246 . . 3 (𝜑𝑆 ∈ 𝒫 𝑋)
7 mreexd.4 . . . . 5 (𝜑𝑌𝑋)
87adantr 480 . . . 4 ((𝜑𝑠 = 𝑆) → 𝑌𝑋)
9 mreexd.5 . . . . . . . 8 (𝜑𝑍 ∈ (𝑁‘(𝑆 ∪ {𝑌})))
109ad2antrr 758 . . . . . . 7 (((𝜑𝑠 = 𝑆) ∧ 𝑦 = 𝑌) → 𝑍 ∈ (𝑁‘(𝑆 ∪ {𝑌})))
11 simplr 788 . . . . . . . . 9 (((𝜑𝑠 = 𝑆) ∧ 𝑦 = 𝑌) → 𝑠 = 𝑆)
12 simpr 476 . . . . . . . . . 10 (((𝜑𝑠 = 𝑆) ∧ 𝑦 = 𝑌) → 𝑦 = 𝑌)
1312sneqd 4137 . . . . . . . . 9 (((𝜑𝑠 = 𝑆) ∧ 𝑦 = 𝑌) → {𝑦} = {𝑌})
1411, 13uneq12d 3730 . . . . . . . 8 (((𝜑𝑠 = 𝑆) ∧ 𝑦 = 𝑌) → (𝑠 ∪ {𝑦}) = (𝑆 ∪ {𝑌}))
1514fveq2d 6092 . . . . . . 7 (((𝜑𝑠 = 𝑆) ∧ 𝑦 = 𝑌) → (𝑁‘(𝑠 ∪ {𝑦})) = (𝑁‘(𝑆 ∪ {𝑌})))
1610, 15eleqtrrd 2691 . . . . . 6 (((𝜑𝑠 = 𝑆) ∧ 𝑦 = 𝑌) → 𝑍 ∈ (𝑁‘(𝑠 ∪ {𝑦})))
17 mreexd.6 . . . . . . . 8 (𝜑 → ¬ 𝑍 ∈ (𝑁𝑆))
1817ad2antrr 758 . . . . . . 7 (((𝜑𝑠 = 𝑆) ∧ 𝑦 = 𝑌) → ¬ 𝑍 ∈ (𝑁𝑆))
1911fveq2d 6092 . . . . . . 7 (((𝜑𝑠 = 𝑆) ∧ 𝑦 = 𝑌) → (𝑁𝑠) = (𝑁𝑆))
2018, 19neleqtrrd 2710 . . . . . 6 (((𝜑𝑠 = 𝑆) ∧ 𝑦 = 𝑌) → ¬ 𝑍 ∈ (𝑁𝑠))
2116, 20eldifd 3551 . . . . 5 (((𝜑𝑠 = 𝑆) ∧ 𝑦 = 𝑌) → 𝑍 ∈ ((𝑁‘(𝑠 ∪ {𝑦})) ∖ (𝑁𝑠)))
22 simplr 788 . . . . . 6 ((((𝜑𝑠 = 𝑆) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) → 𝑦 = 𝑌)
23 simpllr 795 . . . . . . . 8 ((((𝜑𝑠 = 𝑆) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) → 𝑠 = 𝑆)
24 simpr 476 . . . . . . . . 9 ((((𝜑𝑠 = 𝑆) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) → 𝑧 = 𝑍)
2524sneqd 4137 . . . . . . . 8 ((((𝜑𝑠 = 𝑆) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) → {𝑧} = {𝑍})
2623, 25uneq12d 3730 . . . . . . 7 ((((𝜑𝑠 = 𝑆) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) → (𝑠 ∪ {𝑧}) = (𝑆 ∪ {𝑍}))
2726fveq2d 6092 . . . . . 6 ((((𝜑𝑠 = 𝑆) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) → (𝑁‘(𝑠 ∪ {𝑧})) = (𝑁‘(𝑆 ∪ {𝑍})))
2822, 27eleq12d 2682 . . . . 5 ((((𝜑𝑠 = 𝑆) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) → (𝑦 ∈ (𝑁‘(𝑠 ∪ {𝑧})) ↔ 𝑌 ∈ (𝑁‘(𝑆 ∪ {𝑍}))))
2921, 28rspcdv 3285 . . . 4 (((𝜑𝑠 = 𝑆) ∧ 𝑦 = 𝑌) → (∀𝑧 ∈ ((𝑁‘(𝑠 ∪ {𝑦})) ∖ (𝑁𝑠))𝑦 ∈ (𝑁‘(𝑠 ∪ {𝑧})) → 𝑌 ∈ (𝑁‘(𝑆 ∪ {𝑍}))))
308, 29rspcimdv 3283 . . 3 ((𝜑𝑠 = 𝑆) → (∀𝑦𝑋𝑧 ∈ ((𝑁‘(𝑠 ∪ {𝑦})) ∖ (𝑁𝑠))𝑦 ∈ (𝑁‘(𝑠 ∪ {𝑧})) → 𝑌 ∈ (𝑁‘(𝑆 ∪ {𝑍}))))
316, 30rspcimdv 3283 . 2 (𝜑 → (∀𝑠 ∈ 𝒫 𝑋𝑦𝑋𝑧 ∈ ((𝑁‘(𝑠 ∪ {𝑦})) ∖ (𝑁𝑠))𝑦 ∈ (𝑁‘(𝑠 ∪ {𝑧})) → 𝑌 ∈ (𝑁‘(𝑆 ∪ {𝑍}))))
321, 31mpd 15 1 (𝜑𝑌 ∈ (𝑁‘(𝑆 ∪ {𝑍})))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 195  wa 383   = wceq 1475  wcel 1977  wral 2896  cdif 3537  cun 3538  wss 3540  𝒫 cpw 4108  {csn 4125  cfv 5790
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-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4704
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-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  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 4368  df-br 4579  df-iota 5754  df-fv 5798
This theorem is referenced by:  mreexmrid  16075
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