Theorem mrieqvlemd 17596

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 17605 and mrieqv2d 17606. 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 480	. . . 4 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))) → 𝐴 ∈ (Moore‘𝑋))
3		mrieqvlemd.2	. . . 4 ⊢ 𝑁 = (mrCls‘𝐴)
4		undif1 4447	. . . . . 6 ⊢ ((𝑆 ∖ {𝑌}) ∪ {𝑌}) = (𝑆 ∪ {𝑌})
5		mrieqvlemd.3	. . . . . . . . . 10 ⊢ (𝜑 → 𝑆 ⊆ 𝑋)
6	5	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))) → 𝑆 ⊆ 𝑋)
7	6	ssdifssd 4118	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))) → (𝑆 ∖ {𝑌}) ⊆ 𝑋)
8	2, 3, 7	mrcssidd 17592	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))) → (𝑆 ∖ {𝑌}) ⊆ (𝑁‘(𝑆 ∖ {𝑌})))
9		simpr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))) → 𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌})))
10	9	snssd 4781	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))) → {𝑌} ⊆ (𝑁‘(𝑆 ∖ {𝑌})))
11	8, 10	unssd 4163	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))) → ((𝑆 ∖ {𝑌}) ∪ {𝑌}) ⊆ (𝑁‘(𝑆 ∖ {𝑌})))
12	4, 11	eqsstrrid 3994	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))) → (𝑆 ∪ {𝑌}) ⊆ (𝑁‘(𝑆 ∖ {𝑌})))
13	12	unssad 4164	. . . 4 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))) → 𝑆 ⊆ (𝑁‘(𝑆 ∖ {𝑌})))
14		difssd 4108	. . . 4 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))) → (𝑆 ∖ {𝑌}) ⊆ 𝑆)
15	2, 3, 13, 14	mressmrcd 17594	. . 3 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))) → (𝑁‘𝑆) = (𝑁‘(𝑆 ∖ {𝑌})))
16	15	eqcomd 2736	. 2 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))) → (𝑁‘(𝑆 ∖ {𝑌})) = (𝑁‘𝑆))
17	1, 3, 5	mrcssidd 17592	. . . . 5 ⊢ (𝜑 → 𝑆 ⊆ (𝑁‘𝑆))
18		mrieqvlemd.4	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝑆)
19	17, 18	sseldd 3955	. . . 4 ⊢ (𝜑 → 𝑌 ∈ (𝑁‘𝑆))
20	19	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝑁‘(𝑆 ∖ {𝑌})) = (𝑁‘𝑆)) → 𝑌 ∈ (𝑁‘𝑆))
21		simpr 484	. . 3 ⊢ ((𝜑 ∧ (𝑁‘(𝑆 ∖ {𝑌})) = (𝑁‘𝑆)) → (𝑁‘(𝑆 ∖ {𝑌})) = (𝑁‘𝑆))
22	20, 21	eleqtrrd 2832	. 2 ⊢ ((𝜑 ∧ (𝑁‘(𝑆 ∖ {𝑌})) = (𝑁‘𝑆)) → 𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌})))
23	16, 22	impbida 800	1 ⊢ (𝜑 → (𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌})) ↔ (𝑁‘(𝑆 ∖ {𝑌})) = (𝑁‘𝑆)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ∖ cdif 3919   ∪ cun 3920   ⊆ wss 3922  {csn 4597  ‘cfv 6519  Moorecmre 17549  mrClscmrc 17550

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5259  ax-nul 5269  ax-pow 5328  ax-pr 5395  ax-un 7718

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2880  df-ne 2928  df-ral 3047  df-rex 3056  df-rab 3412  df-v 3457  df-sbc 3762  df-csb 3871  df-dif 3925  df-un 3927  df-in 3929  df-ss 3939  df-nul 4305  df-if 4497  df-pw 4573  df-sn 4598  df-pr 4600  df-op 4604  df-uni 4880  df-int 4919  df-br 5116  df-opab 5178  df-mpt 5197  df-id 5541  df-xp 5652  df-rel 5653  df-cnv 5654  df-co 5655  df-dm 5656  df-rn 5657  df-res 5658  df-ima 5659  df-iota 6472  df-fun 6521  df-fn 6522  df-f 6523  df-fv 6527  df-mre 17553  df-mrc 17554

This theorem is referenced by:  mrieqvd  17605  mrieqv2d  17606