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Theorem mreexd 16425
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.1 . . . 4 (𝜑𝑋𝑉)
3 mreexd.3 . . . 4 (𝜑𝑆𝑋)
42, 3sselpwd 4915 . . 3 (𝜑𝑆 ∈ 𝒫 𝑋)
5 mreexd.4 . . . . 5 (𝜑𝑌𝑋)
65adantr 472 . . . 4 ((𝜑𝑠 = 𝑆) → 𝑌𝑋)
7 mreexd.5 . . . . . . . 8 (𝜑𝑍 ∈ (𝑁‘(𝑆 ∪ {𝑌})))
87ad2antrr 764 . . . . . . 7 (((𝜑𝑠 = 𝑆) ∧ 𝑦 = 𝑌) → 𝑍 ∈ (𝑁‘(𝑆 ∪ {𝑌})))
9 simplr 809 . . . . . . . . 9 (((𝜑𝑠 = 𝑆) ∧ 𝑦 = 𝑌) → 𝑠 = 𝑆)
10 simpr 479 . . . . . . . . . 10 (((𝜑𝑠 = 𝑆) ∧ 𝑦 = 𝑌) → 𝑦 = 𝑌)
1110sneqd 4297 . . . . . . . . 9 (((𝜑𝑠 = 𝑆) ∧ 𝑦 = 𝑌) → {𝑦} = {𝑌})
129, 11uneq12d 3876 . . . . . . . 8 (((𝜑𝑠 = 𝑆) ∧ 𝑦 = 𝑌) → (𝑠 ∪ {𝑦}) = (𝑆 ∪ {𝑌}))
1312fveq2d 6308 . . . . . . 7 (((𝜑𝑠 = 𝑆) ∧ 𝑦 = 𝑌) → (𝑁‘(𝑠 ∪ {𝑦})) = (𝑁‘(𝑆 ∪ {𝑌})))
148, 13eleqtrrd 2806 . . . . . 6 (((𝜑𝑠 = 𝑆) ∧ 𝑦 = 𝑌) → 𝑍 ∈ (𝑁‘(𝑠 ∪ {𝑦})))
15 mreexd.6 . . . . . . . 8 (𝜑 → ¬ 𝑍 ∈ (𝑁𝑆))
1615ad2antrr 764 . . . . . . 7 (((𝜑𝑠 = 𝑆) ∧ 𝑦 = 𝑌) → ¬ 𝑍 ∈ (𝑁𝑆))
179fveq2d 6308 . . . . . . 7 (((𝜑𝑠 = 𝑆) ∧ 𝑦 = 𝑌) → (𝑁𝑠) = (𝑁𝑆))
1816, 17neleqtrrd 2825 . . . . . 6 (((𝜑𝑠 = 𝑆) ∧ 𝑦 = 𝑌) → ¬ 𝑍 ∈ (𝑁𝑠))
1914, 18eldifd 3691 . . . . 5 (((𝜑𝑠 = 𝑆) ∧ 𝑦 = 𝑌) → 𝑍 ∈ ((𝑁‘(𝑠 ∪ {𝑦})) ∖ (𝑁𝑠)))
20 simplr 809 . . . . . 6 ((((𝜑𝑠 = 𝑆) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) → 𝑦 = 𝑌)
21 simpllr 817 . . . . . . . 8 ((((𝜑𝑠 = 𝑆) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) → 𝑠 = 𝑆)
22 simpr 479 . . . . . . . . 9 ((((𝜑𝑠 = 𝑆) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) → 𝑧 = 𝑍)
2322sneqd 4297 . . . . . . . 8 ((((𝜑𝑠 = 𝑆) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) → {𝑧} = {𝑍})
2421, 23uneq12d 3876 . . . . . . 7 ((((𝜑𝑠 = 𝑆) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) → (𝑠 ∪ {𝑧}) = (𝑆 ∪ {𝑍}))
2524fveq2d 6308 . . . . . 6 ((((𝜑𝑠 = 𝑆) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) → (𝑁‘(𝑠 ∪ {𝑧})) = (𝑁‘(𝑆 ∪ {𝑍})))
2620, 25eleq12d 2797 . . . . 5 ((((𝜑𝑠 = 𝑆) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) → (𝑦 ∈ (𝑁‘(𝑠 ∪ {𝑧})) ↔ 𝑌 ∈ (𝑁‘(𝑆 ∪ {𝑍}))))
2719, 26rspcdv 3416 . . . 4 (((𝜑𝑠 = 𝑆) ∧ 𝑦 = 𝑌) → (∀𝑧 ∈ ((𝑁‘(𝑠 ∪ {𝑦})) ∖ (𝑁𝑠))𝑦 ∈ (𝑁‘(𝑠 ∪ {𝑧})) → 𝑌 ∈ (𝑁‘(𝑆 ∪ {𝑍}))))
286, 27rspcimdv 3414 . . 3 ((𝜑𝑠 = 𝑆) → (∀𝑦𝑋𝑧 ∈ ((𝑁‘(𝑠 ∪ {𝑦})) ∖ (𝑁𝑠))𝑦 ∈ (𝑁‘(𝑠 ∪ {𝑧})) → 𝑌 ∈ (𝑁‘(𝑆 ∪ {𝑍}))))
294, 28rspcimdv 3414 . 2 (𝜑 → (∀𝑠 ∈ 𝒫 𝑋𝑦𝑋𝑧 ∈ ((𝑁‘(𝑠 ∪ {𝑦})) ∖ (𝑁𝑠))𝑦 ∈ (𝑁‘(𝑠 ∪ {𝑧})) → 𝑌 ∈ (𝑁‘(𝑆 ∪ {𝑍}))))
301, 29mpd 15 1 (𝜑𝑌 ∈ (𝑁‘(𝑆 ∪ {𝑍})))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383   = wceq 1596  wcel 2103  wral 3014  cdif 3677  cun 3678  wss 3680  𝒫 cpw 4266  {csn 4285  cfv 6001
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-sep 4889
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ral 3019  df-rex 3020  df-rab 3023  df-v 3306  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-pw 4268  df-sn 4286  df-pr 4288  df-op 4292  df-uni 4545  df-br 4761  df-iota 5964  df-fv 6009
This theorem is referenced by:  mreexmrid  16426
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