Theorem isacs4lem 18747

Description: In a closure system in which directed unions of closed sets are closed, closure commutes with directed unions. (Contributed by Stefan O'Rear, 2-Apr-2015.)

Hypothesis

Ref	Expression
acsdrscl.f	⊢ 𝐹 = (mrCls‘𝐶)

Assertion

Ref	Expression
isacs4lem	⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → ∪ 𝑠 ∈ 𝐶)) → (𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑡 ∈ 𝒫 𝒫 𝑋((toInc‘𝑡) ∈ Dirset → (𝐹‘∪ 𝑡) = ∪ (𝐹 “ 𝑡))))

Distinct variable groups:   𝐶,𝑠,𝑡   𝐹,𝑠,𝑡   𝑋,𝑠,𝑡

Proof of Theorem isacs4lem

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpll 779	. . . . . . 7 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → ∪ 𝑠 ∈ 𝐶)) ∧ (𝑡 ∈ 𝒫 𝒫 𝑋 ∧ (toInc‘𝑡) ∈ Dirset)) → 𝐶 ∈ (Moore‘𝑋))
2		elpwi 4567	. . . . . . . 8 ⊢ (𝑡 ∈ 𝒫 𝒫 𝑋 → 𝑡 ⊆ 𝒫 𝑋)
3	2	ad2antrl 741	. . . . . . 7 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → ∪ 𝑠 ∈ 𝐶)) ∧ (𝑡 ∈ 𝒫 𝒫 𝑋 ∧ (toInc‘𝑡) ∈ Dirset)) → 𝑡 ⊆ 𝒫 𝑋)
4		acsdrscl.f	. . . . . . . 8 ⊢ 𝐹 = (mrCls‘𝐶)
5	4	mrcuni 17824	. . . . . . 7 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑡 ⊆ 𝒫 𝑋) → (𝐹‘∪ 𝑡) = (𝐹‘∪ (𝐹 “ 𝑡)))
6	1, 3, 5	syl2anc 596	. . . . . 6 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → ∪ 𝑠 ∈ 𝐶)) ∧ (𝑡 ∈ 𝒫 𝒫 𝑋 ∧ (toInc‘𝑡) ∈ Dirset)) → (𝐹‘∪ 𝑡) = (𝐹‘∪ (𝐹 “ 𝑡)))
7	4	mrcf 17812	. . . . . . . . . . . 12 ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝐹:𝒫 𝑋⟶𝐶)
8	7	ffnd 6732	. . . . . . . . . . 11 ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝐹 Fn 𝒫 𝑋)
9	8	adantr 486	. . . . . . . . . 10 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ (𝑡 ∈ 𝒫 𝒫 𝑋 ∧ (toInc‘𝑡) ∈ Dirset)) → 𝐹 Fn 𝒫 𝑋)
10		simpll 779	. . . . . . . . . . 11 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ (𝑡 ∈ 𝒫 𝒫 𝑋 ∧ (toInc‘𝑡) ∈ Dirset)) ∧ (𝑥 ⊆ 𝑦 ∧ 𝑦 ⊆ 𝑋)) → 𝐶 ∈ (Moore‘𝑋))
11		simprl 783	. . . . . . . . . . 11 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ (𝑡 ∈ 𝒫 𝒫 𝑋 ∧ (toInc‘𝑡) ∈ Dirset)) ∧ (𝑥 ⊆ 𝑦 ∧ 𝑦 ⊆ 𝑋)) → 𝑥 ⊆ 𝑦)
12		simprr 785	. . . . . . . . . . 11 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ (𝑡 ∈ 𝒫 𝒫 𝑋 ∧ (toInc‘𝑡) ∈ Dirset)) ∧ (𝑥 ⊆ 𝑦 ∧ 𝑦 ⊆ 𝑋)) → 𝑦 ⊆ 𝑋)
13	10, 4, 11, 12	mrcssd 17827	. . . . . . . . . 10 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ (𝑡 ∈ 𝒫 𝒫 𝑋 ∧ (toInc‘𝑡) ∈ Dirset)) ∧ (𝑥 ⊆ 𝑦 ∧ 𝑦 ⊆ 𝑋)) → (𝐹‘𝑥) ⊆ (𝐹‘𝑦))
14		simprr 785	. . . . . . . . . 10 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ (𝑡 ∈ 𝒫 𝒫 𝑋 ∧ (toInc‘𝑡) ∈ Dirset)) → (toInc‘𝑡) ∈ Dirset)
15	2	ad2antrl 741	. . . . . . . . . 10 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ (𝑡 ∈ 𝒫 𝒫 𝑋 ∧ (toInc‘𝑡) ∈ Dirset)) → 𝑡 ⊆ 𝒫 𝑋)
16	4	fvexi 6922	. . . . . . . . . . . 12 ⊢ 𝐹 ∈ V
17	16	imaex 7950	. . . . . . . . . . 11 ⊢ (𝐹 “ 𝑡) ∈ V
18	17	a1i 11	. . . . . . . . . 10 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ (𝑡 ∈ 𝒫 𝒫 𝑋 ∧ (toInc‘𝑡) ∈ Dirset)) → (𝐹 “ 𝑡) ∈ V)
19	9, 13, 14, 15, 18	ipodrsima 18744	. . . . . . . . 9 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ (𝑡 ∈ 𝒫 𝒫 𝑋 ∧ (toInc‘𝑡) ∈ Dirset)) → (toInc‘(𝐹 “ 𝑡)) ∈ Dirset)
20	19	adantlr 728	. . . . . . . 8 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → ∪ 𝑠 ∈ 𝐶)) ∧ (𝑡 ∈ 𝒫 𝒫 𝑋 ∧ (toInc‘𝑡) ∈ Dirset)) → (toInc‘(𝐹 “ 𝑡)) ∈ Dirset)
21		fveq2 6908	. . . . . . . . . . 11 ⊢ (𝑠 = (𝐹 “ 𝑡) → (toInc‘𝑠) = (toInc‘(𝐹 “ 𝑡)))
22	21	eleq1d 2849	. . . . . . . . . 10 ⊢ (𝑠 = (𝐹 “ 𝑡) → ((toInc‘𝑠) ∈ Dirset ↔ (toInc‘(𝐹 “ 𝑡)) ∈ Dirset))
23		unieq 4881	. . . . . . . . . . 11 ⊢ (𝑠 = (𝐹 “ 𝑡) → ∪ 𝑠 = ∪ (𝐹 “ 𝑡))
24	23	eleq1d 2849	. . . . . . . . . 10 ⊢ (𝑠 = (𝐹 “ 𝑡) → (∪ 𝑠 ∈ 𝐶 ↔ ∪ (𝐹 “ 𝑡) ∈ 𝐶))
25	22, 24	imbi12d 347	. . . . . . . . 9 ⊢ (𝑠 = (𝐹 “ 𝑡) → (((toInc‘𝑠) ∈ Dirset → ∪ 𝑠 ∈ 𝐶) ↔ ((toInc‘(𝐹 “ 𝑡)) ∈ Dirset → ∪ (𝐹 “ 𝑡) ∈ 𝐶)))
26		simplr 781	. . . . . . . . 9 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → ∪ 𝑠 ∈ 𝐶)) ∧ (𝑡 ∈ 𝒫 𝒫 𝑋 ∧ (toInc‘𝑡) ∈ Dirset)) → ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → ∪ 𝑠 ∈ 𝐶))
27		imassrn 6210	. . . . . . . . . . . 12 ⊢ (𝐹 “ 𝑡) ⊆ ran 𝐹
28	7	frnd 6740	. . . . . . . . . . . 12 ⊢ (𝐶 ∈ (Moore‘𝑋) → ran 𝐹 ⊆ 𝐶)
29	27, 28	sstrid 3924	. . . . . . . . . . 11 ⊢ (𝐶 ∈ (Moore‘𝑋) → (𝐹 “ 𝑡) ⊆ 𝐶)
30	17	elpw 4564	. . . . . . . . . . 11 ⊢ ((𝐹 “ 𝑡) ∈ 𝒫 𝐶 ↔ (𝐹 “ 𝑡) ⊆ 𝐶)
31	29, 30	sylibr 237	. . . . . . . . . 10 ⊢ (𝐶 ∈ (Moore‘𝑋) → (𝐹 “ 𝑡) ∈ 𝒫 𝐶)
32	31	ad2antrr 739	. . . . . . . . 9 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → ∪ 𝑠 ∈ 𝐶)) ∧ (𝑡 ∈ 𝒫 𝒫 𝑋 ∧ (toInc‘𝑡) ∈ Dirset)) → (𝐹 “ 𝑡) ∈ 𝒫 𝐶)
33	25, 26, 32	rspcdva 3581	. . . . . . . 8 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → ∪ 𝑠 ∈ 𝐶)) ∧ (𝑡 ∈ 𝒫 𝒫 𝑋 ∧ (toInc‘𝑡) ∈ Dirset)) → ((toInc‘(𝐹 “ 𝑡)) ∈ Dirset → ∪ (𝐹 “ 𝑡) ∈ 𝐶))
34	20, 33	mpd 16	. . . . . . 7 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → ∪ 𝑠 ∈ 𝐶)) ∧ (𝑡 ∈ 𝒫 𝒫 𝑋 ∧ (toInc‘𝑡) ∈ Dirset)) → ∪ (𝐹 “ 𝑡) ∈ 𝐶)
35	4	mrcid 17816	. . . . . . 7 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ ∪ (𝐹 “ 𝑡) ∈ 𝐶) → (𝐹‘∪ (𝐹 “ 𝑡)) = ∪ (𝐹 “ 𝑡))
36	1, 34, 35	syl2anc 596	. . . . . 6 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → ∪ 𝑠 ∈ 𝐶)) ∧ (𝑡 ∈ 𝒫 𝒫 𝑋 ∧ (toInc‘𝑡) ∈ Dirset)) → (𝐹‘∪ (𝐹 “ 𝑡)) = ∪ (𝐹 “ 𝑡))
37	6, 36	eqtrd 2798	. . . . 5 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → ∪ 𝑠 ∈ 𝐶)) ∧ (𝑡 ∈ 𝒫 𝒫 𝑋 ∧ (toInc‘𝑡) ∈ Dirset)) → (𝐹‘∪ 𝑡) = ∪ (𝐹 “ 𝑡))
38	37	exp32 426	. . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → ∪ 𝑠 ∈ 𝐶)) → (𝑡 ∈ 𝒫 𝒫 𝑋 → ((toInc‘𝑡) ∈ Dirset → (𝐹‘∪ 𝑡) = ∪ (𝐹 “ 𝑡))))
39	38	ralrimiv 3157	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → ∪ 𝑠 ∈ 𝐶)) → ∀𝑡 ∈ 𝒫 𝒫 𝑋((toInc‘𝑡) ∈ Dirset → (𝐹‘∪ 𝑡) = ∪ (𝐹 “ 𝑡)))
40	39	ex 418	. 2 ⊢ (𝐶 ∈ (Moore‘𝑋) → (∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → ∪ 𝑠 ∈ 𝐶) → ∀𝑡 ∈ 𝒫 𝒫 𝑋((toInc‘𝑡) ∈ Dirset → (𝐹‘∪ 𝑡) = ∪ (𝐹 “ 𝑡))))
41	40	imdistani 579	1 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → ∪ 𝑠 ∈ 𝐶)) → (𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑡 ∈ 𝒫 𝒫 𝑋((toInc‘𝑡) ∈ Dirset → (𝐹‘∪ 𝑡) = ∪ (𝐹 “ 𝑡))))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∀wral 3080  Vcvv 3454   ⊆ wss 3900  𝒫 cpw 4560  ∪ cuni 4870  ran crn 5656   “ cima 5658   Fn wfn 6553  ‘cfv 6558  Moorecmre 17781  mrClscmrc 17782  Dirsetcdrs 18496  toInccipo 18730

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775  ax-cnex 11283  ax-resscn 11284  ax-1cn 11285  ax-icn 11286  ax-addcl 11287  ax-addrcl 11288  ax-mulcl 11289  ax-mulrcl 11290  ax-mulcom 11291  ax-addass 11292  ax-mulass 11293  ax-distr 11294  ax-i2m1 11295  ax-1ne0 11296  ax-1rid 11297  ax-rnegex 11298  ax-rrecex 11299  ax-cnre 11300  ax-pre-lttri 11301  ax-pre-lttrn 11302  ax-pre-ltadd 11303  ax-pre-mulgt0 11304

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6322  df-ord 6383  df-on 6384  df-lim 6385  df-suc 6386  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-riota 7400  df-ov 7446  df-oprab 7447  df-mpo 7448  df-om 7902  df-1st 8025  df-2nd 8026  df-frecs 8327  df-wrecs 8358  df-recs 8407  df-rdg 8446  df-1o 8504  df-er 8745  df-en 9002  df-dom 9003  df-sdom 9004  df-fin 9005  df-pnf 11372  df-mnf 11373  df-xr 11374  df-ltxr 11375  df-le 11376  df-sub 11570  df-neg 11571  df-nn 12363  df-2 12432  df-3 12433  df-4 12434  df-5 12435  df-6 12436  df-7 12437  df-8 12438  df-9 12439  df-n0 12634  df-z 12721  df-dec 12842  df-uz 12993  df-fz 13667  df-struct 17354  df-slot 17389  df-ndx 17401  df-base 17417  df-tset 17476  df-ple 17477  df-ocomp 17478  df-mre 17785  df-mrc 17786  df-proset 18497  df-drs 18498  df-poset 18516  df-ipo 18731

This theorem is used by:  acsdrscl  18749  acsficl  18750  isacs5  18751  isacs4  18752  isacs3  18753