Theorem mreexdomd 17555

Description: In a Moore system whose closure operator has the exchange property, if 𝑆 is independent and contained in the closure of 𝑇, and either 𝑆 or 𝑇 is finite, then 𝑇 dominates 𝑆. This is an immediate consequence of mreexexd 17554. (Contributed by David Moews, 1-May-2017.)

Hypotheses

Ref	Expression
mreexdomd.1	⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋))
mreexdomd.2	⊢ 𝑁 = (mrCls‘𝐴)
mreexdomd.3	⊢ 𝐼 = (mrInd‘𝐴)
mreexdomd.4	⊢ (𝜑 → ∀𝑠 ∈ 𝒫 𝑋∀𝑦 ∈ 𝑋 ∀𝑧 ∈ ((𝑁‘(𝑠 ∪ {𝑦})) ∖ (𝑁‘𝑠))𝑦 ∈ (𝑁‘(𝑠 ∪ {𝑧})))
mreexdomd.5	⊢ (𝜑 → 𝑆 ⊆ (𝑁‘𝑇))
mreexdomd.6	⊢ (𝜑 → 𝑇 ⊆ 𝑋)
mreexdomd.7	⊢ (𝜑 → (𝑆 ∈ Fin ∨ 𝑇 ∈ Fin))
mreexdomd.8	⊢ (𝜑 → 𝑆 ∈ 𝐼)

Assertion

Ref	Expression
mreexdomd	⊢ (𝜑 → 𝑆 ≼ 𝑇)

Distinct variable groups:   𝑋,𝑠,𝑦,𝑧   𝜑,𝑠,𝑦,𝑧   𝐼,𝑠,𝑦,𝑧   𝑁,𝑠,𝑦,𝑧

Allowed substitution hints:   𝐴(𝑦,𝑧,𝑠)   𝑆(𝑦,𝑧,𝑠)   𝑇(𝑦,𝑧,𝑠)

Proof of Theorem mreexdomd

Dummy variable 𝑖 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mreexdomd.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋))
2		mreexdomd.2	. . 3 ⊢ 𝑁 = (mrCls‘𝐴)
3		mreexdomd.3	. . 3 ⊢ 𝐼 = (mrInd‘𝐴)
4		mreexdomd.4	. . 3 ⊢ (𝜑 → ∀𝑠 ∈ 𝒫 𝑋∀𝑦 ∈ 𝑋 ∀𝑧 ∈ ((𝑁‘(𝑠 ∪ {𝑦})) ∖ (𝑁‘𝑠))𝑦 ∈ (𝑁‘(𝑠 ∪ {𝑧})))
5		mreexdomd.8	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ 𝐼)
6	3, 1, 5	mrissd 17542	. . . 4 ⊢ (𝜑 → 𝑆 ⊆ 𝑋)
7		dif0 4329	. . . 4 ⊢ (𝑋 ∖ ∅) = 𝑋
8	6, 7	sseqtrrdi 3977	. . 3 ⊢ (𝜑 → 𝑆 ⊆ (𝑋 ∖ ∅))
9		mreexdomd.6	. . . 4 ⊢ (𝜑 → 𝑇 ⊆ 𝑋)
10	9, 7	sseqtrrdi 3977	. . 3 ⊢ (𝜑 → 𝑇 ⊆ (𝑋 ∖ ∅))
11		mreexdomd.5	. . . 4 ⊢ (𝜑 → 𝑆 ⊆ (𝑁‘𝑇))
12		un0 4345	. . . . 5 ⊢ (𝑇 ∪ ∅) = 𝑇
13	12	fveq2i 6825	. . . 4 ⊢ (𝑁‘(𝑇 ∪ ∅)) = (𝑁‘𝑇)
14	11, 13	sseqtrrdi 3977	. . 3 ⊢ (𝜑 → 𝑆 ⊆ (𝑁‘(𝑇 ∪ ∅)))
15		un0 4345	. . . 4 ⊢ (𝑆 ∪ ∅) = 𝑆
16	15, 5	eqeltrid 2832	. . 3 ⊢ (𝜑 → (𝑆 ∪ ∅) ∈ 𝐼)
17		mreexdomd.7	. . 3 ⊢ (𝜑 → (𝑆 ∈ Fin ∨ 𝑇 ∈ Fin))
18	1, 2, 3, 4, 8, 10, 14, 16, 17	mreexexd 17554	. 2 ⊢ (𝜑 → ∃𝑖 ∈ 𝒫 𝑇(𝑆 ≈ 𝑖 ∧ (𝑖 ∪ ∅) ∈ 𝐼))
19		simprrl 780	. . 3 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝒫 𝑇 ∧ (𝑆 ≈ 𝑖 ∧ (𝑖 ∪ ∅) ∈ 𝐼))) → 𝑆 ≈ 𝑖)
20		simprl 770	. . . . 5 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝒫 𝑇 ∧ (𝑆 ≈ 𝑖 ∧ (𝑖 ∪ ∅) ∈ 𝐼))) → 𝑖 ∈ 𝒫 𝑇)
21	20	elpwid 4560	. . . 4 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝒫 𝑇 ∧ (𝑆 ≈ 𝑖 ∧ (𝑖 ∪ ∅) ∈ 𝐼))) → 𝑖 ⊆ 𝑇)
22	1	elfvexd 6859	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ V)
23	22, 9	ssexd 5263	. . . . . 6 ⊢ (𝜑 → 𝑇 ∈ V)
24		ssdomg 8925	. . . . . 6 ⊢ (𝑇 ∈ V → (𝑖 ⊆ 𝑇 → 𝑖 ≼ 𝑇))
25	23, 24	syl 17	. . . . 5 ⊢ (𝜑 → (𝑖 ⊆ 𝑇 → 𝑖 ≼ 𝑇))
26	25	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝒫 𝑇 ∧ (𝑆 ≈ 𝑖 ∧ (𝑖 ∪ ∅) ∈ 𝐼))) → (𝑖 ⊆ 𝑇 → 𝑖 ≼ 𝑇))
27	21, 26	mpd 15	. . 3 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝒫 𝑇 ∧ (𝑆 ≈ 𝑖 ∧ (𝑖 ∪ ∅) ∈ 𝐼))) → 𝑖 ≼ 𝑇)
28		endomtr 8937	. . 3 ⊢ ((𝑆 ≈ 𝑖 ∧ 𝑖 ≼ 𝑇) → 𝑆 ≼ 𝑇)
29	19, 27, 28	syl2anc 584	. 2 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝒫 𝑇 ∧ (𝑆 ≈ 𝑖 ∧ (𝑖 ∪ ∅) ∈ 𝐼))) → 𝑆 ≼ 𝑇)
30	18, 29	rexlimddv 3136	1 ⊢ (𝜑 → 𝑆 ≼ 𝑇)

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 847   = wceq 1540   ∈ wcel 2109  ∀wral 3044  Vcvv 3436   ∖ cdif 3900   ∪ cun 3901   ⊆ wss 3903  ∅c0 4284  𝒫 cpw 4551  {csn 4577   class class class wbr 5092  ‘cfv 6482   ≈ cen 8869   ≼ cdom 8870  Fincfn 8872  Moorecmre 17484  mrClscmrc 17485  mrIndcmri 17486

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 2701  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-int 4897  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-om 7800  df-1o 8388  df-er 8625  df-en 8873  df-dom 8874  df-sdom 8875  df-fin 8876  df-card 9835  df-mre 17488  df-mrc 17489  df-mri 17490

This theorem is referenced by:  mreexfidimd  17556