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 4325	. . . 4 ⊢ (𝑋 ∖ ∅) = 𝑋
8	6, 7	sseqtrrdi 3971	. . 3 ⊢ (𝜑 → 𝑆 ⊆ (𝑋 ∖ ∅))
9		mreexdomd.6	. . . 4 ⊢ (𝜑 → 𝑇 ⊆ 𝑋)
10	9, 7	sseqtrrdi 3971	. . 3 ⊢ (𝜑 → 𝑇 ⊆ (𝑋 ∖ ∅))
11		mreexdomd.5	. . . 4 ⊢ (𝜑 → 𝑆 ⊆ (𝑁‘𝑇))
12		un0 4341	. . . . 5 ⊢ (𝑇 ∪ ∅) = 𝑇
13	12	fveq2i 6825	. . . 4 ⊢ (𝑁‘(𝑇 ∪ ∅)) = (𝑁‘𝑇)
14	11, 13	sseqtrrdi 3971	. . 3 ⊢ (𝜑 → 𝑆 ⊆ (𝑁‘(𝑇 ∪ ∅)))
15		un0 4341	. . . 4 ⊢ (𝑆 ∪ ∅) = 𝑆
16	15, 5	eqeltrid 2835	. . 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 4556	. . . 4 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝒫 𝑇 ∧ (𝑆 ≈ 𝑖 ∧ (𝑖 ∪ ∅) ∈ 𝐼))) → 𝑖 ⊆ 𝑇)
22	1	elfvexd 6858	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ V)
23	22, 9	ssexd 5260	. . . . . 6 ⊢ (𝜑 → 𝑇 ∈ V)
24		ssdomg 8922	. . . . . 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 8934	. . 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 3139	1 ⊢ (𝜑 → 𝑆 ≼ 𝑇)

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 847   = wceq 1541   ∈ wcel 2111  ∀wral 3047  Vcvv 3436   ∖ cdif 3894   ∪ cun 3895   ⊆ wss 3897  ∅c0 4280  𝒫 cpw 4547  {csn 4573   class class class wbr 5089  ‘cfv 6481   ≈ cen 8866   ≼ cdom 8867  Fincfn 8869  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 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-int 4896  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-om 7797  df-1o 8385  df-er 8622  df-en 8870  df-dom 8871  df-sdom 8872  df-fin 8873  df-card 9832  df-mre 17488  df-mrc 17489  df-mri 17490

This theorem is referenced by:  mreexfidimd  17556