Theorem isacs4lem 18510

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 767	. . . . . . 7 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → ∪ 𝑠 ∈ 𝐶)) ∧ (𝑡 ∈ 𝒫 𝒫 𝑋 ∧ (toInc‘𝑡) ∈ Dirset)) → 𝐶 ∈ (Moore‘𝑋))
2		elpwi 4548	. . . . . . . 8 ⊢ (𝑡 ∈ 𝒫 𝒫 𝑋 → 𝑡 ⊆ 𝒫 𝑋)
3	2	ad2antrl 729	. . . . . . 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 17587	. . . . . . 7 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑡 ⊆ 𝒫 𝑋) → (𝐹‘∪ 𝑡) = (𝐹‘∪ (𝐹 “ 𝑡)))
6	1, 3, 5	syl2anc 585	. . . . . 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 17575	. . . . . . . . . . . 12 ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝐹:𝒫 𝑋⟶𝐶)
8	7	ffnd 6669	. . . . . . . . . . 11 ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝐹 Fn 𝒫 𝑋)
9	8	adantr 480	. . . . . . . . . 10 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ (𝑡 ∈ 𝒫 𝒫 𝑋 ∧ (toInc‘𝑡) ∈ Dirset)) → 𝐹 Fn 𝒫 𝑋)
10		simpll 767	. . . . . . . . . . 11 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ (𝑡 ∈ 𝒫 𝒫 𝑋 ∧ (toInc‘𝑡) ∈ Dirset)) ∧ (𝑥 ⊆ 𝑦 ∧ 𝑦 ⊆ 𝑋)) → 𝐶 ∈ (Moore‘𝑋))
11		simprl 771	. . . . . . . . . . 11 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ (𝑡 ∈ 𝒫 𝒫 𝑋 ∧ (toInc‘𝑡) ∈ Dirset)) ∧ (𝑥 ⊆ 𝑦 ∧ 𝑦 ⊆ 𝑋)) → 𝑥 ⊆ 𝑦)
12		simprr 773	. . . . . . . . . . 11 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ (𝑡 ∈ 𝒫 𝒫 𝑋 ∧ (toInc‘𝑡) ∈ Dirset)) ∧ (𝑥 ⊆ 𝑦 ∧ 𝑦 ⊆ 𝑋)) → 𝑦 ⊆ 𝑋)
13	10, 4, 11, 12	mrcssd 17590	. . . . . . . . . 10 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ (𝑡 ∈ 𝒫 𝒫 𝑋 ∧ (toInc‘𝑡) ∈ Dirset)) ∧ (𝑥 ⊆ 𝑦 ∧ 𝑦 ⊆ 𝑋)) → (𝐹‘𝑥) ⊆ (𝐹‘𝑦))
14		simprr 773	. . . . . . . . . 10 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ (𝑡 ∈ 𝒫 𝒫 𝑋 ∧ (toInc‘𝑡) ∈ Dirset)) → (toInc‘𝑡) ∈ Dirset)
15	2	ad2antrl 729	. . . . . . . . . 10 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ (𝑡 ∈ 𝒫 𝒫 𝑋 ∧ (toInc‘𝑡) ∈ Dirset)) → 𝑡 ⊆ 𝒫 𝑋)
16	4	fvexi 6854	. . . . . . . . . . . 12 ⊢ 𝐹 ∈ V
17	16	imaex 7865	. . . . . . . . . . 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 18507	. . . . . . . . 9 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ (𝑡 ∈ 𝒫 𝒫 𝑋 ∧ (toInc‘𝑡) ∈ Dirset)) → (toInc‘(𝐹 “ 𝑡)) ∈ Dirset)
20	19	adantlr 716	. . . . . . . 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 6840	. . . . . . . . . . 11 ⊢ (𝑠 = (𝐹 “ 𝑡) → (toInc‘𝑠) = (toInc‘(𝐹 “ 𝑡)))
22	21	eleq1d 2821	. . . . . . . . . 10 ⊢ (𝑠 = (𝐹 “ 𝑡) → ((toInc‘𝑠) ∈ Dirset ↔ (toInc‘(𝐹 “ 𝑡)) ∈ Dirset))
23		unieq 4861	. . . . . . . . . . 11 ⊢ (𝑠 = (𝐹 “ 𝑡) → ∪ 𝑠 = ∪ (𝐹 “ 𝑡))
24	23	eleq1d 2821	. . . . . . . . . 10 ⊢ (𝑠 = (𝐹 “ 𝑡) → (∪ 𝑠 ∈ 𝐶 ↔ ∪ (𝐹 “ 𝑡) ∈ 𝐶))
25	22, 24	imbi12d 344	. . . . . . . . 9 ⊢ (𝑠 = (𝐹 “ 𝑡) → (((toInc‘𝑠) ∈ Dirset → ∪ 𝑠 ∈ 𝐶) ↔ ((toInc‘(𝐹 “ 𝑡)) ∈ Dirset → ∪ (𝐹 “ 𝑡) ∈ 𝐶)))
26		simplr 769	. . . . . . . . 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 6036	. . . . . . . . . . . 12 ⊢ (𝐹 “ 𝑡) ⊆ ran 𝐹
28	7	frnd 6676	. . . . . . . . . . . 12 ⊢ (𝐶 ∈ (Moore‘𝑋) → ran 𝐹 ⊆ 𝐶)
29	27, 28	sstrid 3933	. . . . . . . . . . 11 ⊢ (𝐶 ∈ (Moore‘𝑋) → (𝐹 “ 𝑡) ⊆ 𝐶)
30	17	elpw 4545	. . . . . . . . . . 11 ⊢ ((𝐹 “ 𝑡) ∈ 𝒫 𝐶 ↔ (𝐹 “ 𝑡) ⊆ 𝐶)
31	29, 30	sylibr 234	. . . . . . . . . 10 ⊢ (𝐶 ∈ (Moore‘𝑋) → (𝐹 “ 𝑡) ∈ 𝒫 𝐶)
32	31	ad2antrr 727	. . . . . . . . 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 3565	. . . . . . . 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 15	. . . . . . 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 17579	. . . . . . 7 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ ∪ (𝐹 “ 𝑡) ∈ 𝐶) → (𝐹‘∪ (𝐹 “ 𝑡)) = ∪ (𝐹 “ 𝑡))
36	1, 34, 35	syl2anc 585	. . . . . 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 2771	. . . . 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 420	. . . 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 3128	. . 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 412	. 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 568	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))))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3051  Vcvv 3429   ⊆ wss 3889  𝒫 cpw 4541  ∪ cuni 4850  ran crn 5632   “ cima 5634   Fn wfn 6493  ‘cfv 6498  Moorecmre 17544  mrClscmrc 17545  Dirsetcdrs 18259  toInccipo 18493

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-int 4890  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  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-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-1st 7942  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-1o 8405  df-er 8643  df-en 8894  df-dom 8895  df-sdom 8896  df-fin 8897  df-pnf 11181  df-mnf 11182  df-xr 11183  df-ltxr 11184  df-le 11185  df-sub 11379  df-neg 11380  df-nn 12175  df-2 12244  df-3 12245  df-4 12246  df-5 12247  df-6 12248  df-7 12249  df-8 12250  df-9 12251  df-n0 12438  df-z 12525  df-dec 12645  df-uz 12789  df-fz 13462  df-struct 17117  df-slot 17152  df-ndx 17164  df-base 17180  df-tset 17239  df-ple 17240  df-ocomp 17241  df-mre 17548  df-mrc 17549  df-proset 18260  df-drs 18261  df-poset 18279  df-ipo 18494

This theorem is referenced by:  acsdrscl  18512  acsficl  18513  isacs5  18514  isacs4  18515  isacs3  18516