Theorem mrieqvlemd 17723

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 17732 and mrieqv2d 17733. Deduction form. (Contributed by David Moews, 1-May-2017.)

Hypotheses

Ref	Expression
mrieqvlemd.1	⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋))
mrieqvlemd.2	⊢ 𝑁 = (mrCls‘𝐴)
mrieqvlemd.3	⊢ (𝜑 → 𝑆 ⊆ 𝑋)
mrieqvlemd.4	⊢ (𝜑 → 𝑌 ∈ 𝑆)

Assertion

Ref	Expression
mrieqvlemd	⊢ (𝜑 → (𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌})) ↔ (𝑁‘(𝑆 ∖ {𝑌})) = (𝑁‘𝑆)))

Proof of Theorem mrieqvlemd

Step	Hyp	Ref	Expression
1		mrieqvlemd.1	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋))
2	1	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))) → 𝐴 ∈ (Moore‘𝑋))
3		mrieqvlemd.2	. . . 4 ⊢ 𝑁 = (mrCls‘𝐴)
4		undif1 4433	. . . . . 6 ⊢ ((𝑆 ∖ {𝑌}) ∪ {𝑌}) = (𝑆 ∪ {𝑌})
5		mrieqvlemd.3	. . . . . . . . . 10 ⊢ (𝜑 → 𝑆 ⊆ 𝑋)
6	5	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))) → 𝑆 ⊆ 𝑋)
7	6	ssdifssd 4097	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))) → (𝑆 ∖ {𝑌}) ⊆ 𝑋)
8	2, 3, 7	mrcssidd 17719	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))) → (𝑆 ∖ {𝑌}) ⊆ (𝑁‘(𝑆 ∖ {𝑌})))
9		simpr 490	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))) → 𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌})))
10	9	snssd 4750	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))) → {𝑌} ⊆ (𝑁‘(𝑆 ∖ {𝑌})))
11	8, 10	unssd 4141	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))) → ((𝑆 ∖ {𝑌}) ∪ {𝑌}) ⊆ (𝑁‘(𝑆 ∖ {𝑌})))
12	4, 11	eqsstrrid 3973	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))) → (𝑆 ∪ {𝑌}) ⊆ (𝑁‘(𝑆 ∖ {𝑌})))
13	12	unssad 4142	. . . 4 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))) → 𝑆 ⊆ (𝑁‘(𝑆 ∖ {𝑌})))
14		difssd 4087	. . . 4 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))) → (𝑆 ∖ {𝑌}) ⊆ 𝑆)
15	2, 3, 13, 14	mressmrcd 17721	. . 3 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))) → (𝑁‘𝑆) = (𝑁‘(𝑆 ∖ {𝑌})))
16	15	eqcomd 2768	. 2 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))) → (𝑁‘(𝑆 ∖ {𝑌})) = (𝑁‘𝑆))
17	1, 3, 5	mrcssidd 17719	. . . . 5 ⊢ (𝜑 → 𝑆 ⊆ (𝑁‘𝑆))
18		mrieqvlemd.4	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝑆)
19	17, 18	sseldd 3935	. . . 4 ⊢ (𝜑 → 𝑌 ∈ (𝑁‘𝑆))
20	19	adantr 486	. . 3 ⊢ ((𝜑 ∧ (𝑁‘(𝑆 ∖ {𝑌})) = (𝑁‘𝑆)) → 𝑌 ∈ (𝑁‘𝑆))
21		simpr 490	. . 3 ⊢ ((𝜑 ∧ (𝑁‘(𝑆 ∖ {𝑌})) = (𝑁‘𝑆)) → (𝑁‘(𝑆 ∖ {𝑌})) = (𝑁‘𝑆))
22	20, 21	eleqtrrd 2865	. 2 ⊢ ((𝜑 ∧ (𝑁‘(𝑆 ∖ {𝑌})) = (𝑁‘𝑆)) → 𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌})))
23	16, 22	impbida 813	1 ⊢ (𝜑 → (𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌})) ↔ (𝑁‘(𝑆 ∖ {𝑌})) = (𝑁‘𝑆)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145   ∖ cdif 3899   ∪ cun 3900   ⊆ wss 3902  {csn 4587  ‘cfv 6537  Moorecmre 17672  mrClscmrc 17673

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7740

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-int 4911  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-fv 6545  df-mre 17676  df-mrc 17677

This theorem is used by:  mrieqvd  17732  mrieqv2d  17733