Theorem mreexdomd 17586

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 17585. (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 17573	. . . 4 ⊢ (𝜑 → 𝑆 ⊆ 𝑋)
7		dif0 4337	. . . 4 ⊢ (𝑋 ∖ ∅) = 𝑋
8	6, 7	sseqtrrdi 3985	. . 3 ⊢ (𝜑 → 𝑆 ⊆ (𝑋 ∖ ∅))
9		mreexdomd.6	. . . 4 ⊢ (𝜑 → 𝑇 ⊆ 𝑋)
10	9, 7	sseqtrrdi 3985	. . 3 ⊢ (𝜑 → 𝑇 ⊆ (𝑋 ∖ ∅))
11		mreexdomd.5	. . . 4 ⊢ (𝜑 → 𝑆 ⊆ (𝑁‘𝑇))
12		un0 4353	. . . . 5 ⊢ (𝑇 ∪ ∅) = 𝑇
13	12	fveq2i 6843	. . . 4 ⊢ (𝑁‘(𝑇 ∪ ∅)) = (𝑁‘𝑇)
14	11, 13	sseqtrrdi 3985	. . 3 ⊢ (𝜑 → 𝑆 ⊆ (𝑁‘(𝑇 ∪ ∅)))
15		un0 4353	. . . 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 17585	. 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 4568	. . . 4 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝒫 𝑇 ∧ (𝑆 ≈ 𝑖 ∧ (𝑖 ∪ ∅) ∈ 𝐼))) → 𝑖 ⊆ 𝑇)
22	1	elfvexd 6879	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ V)
23	22, 9	ssexd 5274	. . . . . 6 ⊢ (𝜑 → 𝑇 ∈ V)
24		ssdomg 8948	. . . . . 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 8960	. . 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 3140	1 ⊢ (𝜑 → 𝑆 ≼ 𝑇)

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 847   = wceq 1540   ∈ wcel 2109  ∀wral 3044  Vcvv 3444   ∖ cdif 3908   ∪ cun 3909   ⊆ wss 3911  ∅c0 4292  𝒫 cpw 4559  {csn 4585   class class class wbr 5102  ‘cfv 6499   ≈ cen 8892   ≼ cdom 8893  Fincfn 8895  Moorecmre 17519  mrClscmrc 17520  mrIndcmri 17521

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 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691

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 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-int 4907  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-om 7823  df-1o 8411  df-er 8648  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-card 9868  df-mre 17523  df-mrc 17524  df-mri 17525

This theorem is referenced by:  mreexfidimd  17587