Theorem mreexdomd 17707

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 17706. (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 17694	. . . 4 ⊢ (𝜑 → 𝑆 ⊆ 𝑋)
7		dif0 4400	. . . 4 ⊢ (𝑋 ∖ ∅) = 𝑋
8	6, 7	sseqtrrdi 4060	. . 3 ⊢ (𝜑 → 𝑆 ⊆ (𝑋 ∖ ∅))
9		mreexdomd.6	. . . 4 ⊢ (𝜑 → 𝑇 ⊆ 𝑋)
10	9, 7	sseqtrrdi 4060	. . 3 ⊢ (𝜑 → 𝑇 ⊆ (𝑋 ∖ ∅))
11		mreexdomd.5	. . . 4 ⊢ (𝜑 → 𝑆 ⊆ (𝑁‘𝑇))
12		un0 4417	. . . . 5 ⊢ (𝑇 ∪ ∅) = 𝑇
13	12	fveq2i 6923	. . . 4 ⊢ (𝑁‘(𝑇 ∪ ∅)) = (𝑁‘𝑇)
14	11, 13	sseqtrrdi 4060	. . 3 ⊢ (𝜑 → 𝑆 ⊆ (𝑁‘(𝑇 ∪ ∅)))
15		un0 4417	. . . 4 ⊢ (𝑆 ∪ ∅) = 𝑆
16	15, 5	eqeltrid 2848	. . 3 ⊢ (𝜑 → (𝑆 ∪ ∅) ∈ 𝐼)
17		mreexdomd.7	. . 3 ⊢ (𝜑 → (𝑆 ∈ Fin ∨ 𝑇 ∈ Fin))
18	1, 2, 3, 4, 8, 10, 14, 16, 17	mreexexd 17706	. 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 4631	. . . 4 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝒫 𝑇 ∧ (𝑆 ≈ 𝑖 ∧ (𝑖 ∪ ∅) ∈ 𝐼))) → 𝑖 ⊆ 𝑇)
22	1	elfvexd 6959	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ V)
23	22, 9	ssexd 5342	. . . . . 6 ⊢ (𝜑 → 𝑇 ∈ V)
24		ssdomg 9060	. . . . . 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 9072	. . 3 ⊢ ((𝑆 ≈ 𝑖 ∧ 𝑖 ≼ 𝑇) → 𝑆 ≼ 𝑇)
29	19, 27, 28	syl2anc 583	. 2 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝒫 𝑇 ∧ (𝑆 ≈ 𝑖 ∧ (𝑖 ∪ ∅) ∈ 𝐼))) → 𝑆 ≼ 𝑇)
30	18, 29	rexlimddv 3167	1 ⊢ (𝜑 → 𝑆 ≼ 𝑇)

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 846   = wceq 1537   ∈ wcel 2108  ∀wral 3067  Vcvv 3488   ∖ cdif 3973   ∪ cun 3974   ⊆ wss 3976  ∅c0 4352  𝒫 cpw 4622  {csn 4648   class class class wbr 5166  ‘cfv 6573   ≈ cen 9000   ≼ cdom 9001  Fincfn 9003  Moorecmre 17640  mrClscmrc 17641  mrIndcmri 17642

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-om 7904  df-1o 8522  df-er 8763  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-card 10008  df-mre 17644  df-mrc 17645  df-mri 17646

This theorem is referenced by:  mreexfidimd  17708