Theorem isacs4lem 18554

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 766	. . . . . . 7 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → ∪ 𝑠 ∈ 𝐶)) ∧ (𝑡 ∈ 𝒫 𝒫 𝑋 ∧ (toInc‘𝑡) ∈ Dirset)) → 𝐶 ∈ (Moore‘𝑋))
2		elpwi 4582	. . . . . . . 8 ⊢ (𝑡 ∈ 𝒫 𝒫 𝑋 → 𝑡 ⊆ 𝒫 𝑋)
3	2	ad2antrl 728	. . . . . . 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 17633	. . . . . . 7 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑡 ⊆ 𝒫 𝑋) → (𝐹‘∪ 𝑡) = (𝐹‘∪ (𝐹 “ 𝑡)))
6	1, 3, 5	syl2anc 584	. . . . . 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 17621	. . . . . . . . . . . 12 ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝐹:𝒫 𝑋⟶𝐶)
8	7	ffnd 6707	. . . . . . . . . . 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 766	. . . . . . . . . . 11 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ (𝑡 ∈ 𝒫 𝒫 𝑋 ∧ (toInc‘𝑡) ∈ Dirset)) ∧ (𝑥 ⊆ 𝑦 ∧ 𝑦 ⊆ 𝑋)) → 𝐶 ∈ (Moore‘𝑋))
11		simprl 770	. . . . . . . . . . 11 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ (𝑡 ∈ 𝒫 𝒫 𝑋 ∧ (toInc‘𝑡) ∈ Dirset)) ∧ (𝑥 ⊆ 𝑦 ∧ 𝑦 ⊆ 𝑋)) → 𝑥 ⊆ 𝑦)
12		simprr 772	. . . . . . . . . . 11 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ (𝑡 ∈ 𝒫 𝒫 𝑋 ∧ (toInc‘𝑡) ∈ Dirset)) ∧ (𝑥 ⊆ 𝑦 ∧ 𝑦 ⊆ 𝑋)) → 𝑦 ⊆ 𝑋)
13	10, 4, 11, 12	mrcssd 17636	. . . . . . . . . 10 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ (𝑡 ∈ 𝒫 𝒫 𝑋 ∧ (toInc‘𝑡) ∈ Dirset)) ∧ (𝑥 ⊆ 𝑦 ∧ 𝑦 ⊆ 𝑋)) → (𝐹‘𝑥) ⊆ (𝐹‘𝑦))
14		simprr 772	. . . . . . . . . 10 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ (𝑡 ∈ 𝒫 𝒫 𝑋 ∧ (toInc‘𝑡) ∈ Dirset)) → (toInc‘𝑡) ∈ Dirset)
15	2	ad2antrl 728	. . . . . . . . . 10 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ (𝑡 ∈ 𝒫 𝒫 𝑋 ∧ (toInc‘𝑡) ∈ Dirset)) → 𝑡 ⊆ 𝒫 𝑋)
16	4	fvexi 6890	. . . . . . . . . . . 12 ⊢ 𝐹 ∈ V
17	16	imaex 7910	. . . . . . . . . . 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 18551	. . . . . . . . 9 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ (𝑡 ∈ 𝒫 𝒫 𝑋 ∧ (toInc‘𝑡) ∈ Dirset)) → (toInc‘(𝐹 “ 𝑡)) ∈ Dirset)
20	19	adantlr 715	. . . . . . . 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 6876	. . . . . . . . . . 11 ⊢ (𝑠 = (𝐹 “ 𝑡) → (toInc‘𝑠) = (toInc‘(𝐹 “ 𝑡)))
22	21	eleq1d 2819	. . . . . . . . . 10 ⊢ (𝑠 = (𝐹 “ 𝑡) → ((toInc‘𝑠) ∈ Dirset ↔ (toInc‘(𝐹 “ 𝑡)) ∈ Dirset))
23		unieq 4894	. . . . . . . . . . 11 ⊢ (𝑠 = (𝐹 “ 𝑡) → ∪ 𝑠 = ∪ (𝐹 “ 𝑡))
24	23	eleq1d 2819	. . . . . . . . . 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 768	. . . . . . . . 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 6058	. . . . . . . . . . . 12 ⊢ (𝐹 “ 𝑡) ⊆ ran 𝐹
28	7	frnd 6714	. . . . . . . . . . . 12 ⊢ (𝐶 ∈ (Moore‘𝑋) → ran 𝐹 ⊆ 𝐶)
29	27, 28	sstrid 3970	. . . . . . . . . . 11 ⊢ (𝐶 ∈ (Moore‘𝑋) → (𝐹 “ 𝑡) ⊆ 𝐶)
30	17	elpw 4579	. . . . . . . . . . 11 ⊢ ((𝐹 “ 𝑡) ∈ 𝒫 𝐶 ↔ (𝐹 “ 𝑡) ⊆ 𝐶)
31	29, 30	sylibr 234	. . . . . . . . . 10 ⊢ (𝐶 ∈ (Moore‘𝑋) → (𝐹 “ 𝑡) ∈ 𝒫 𝐶)
32	31	ad2antrr 726	. . . . . . . . 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 3602	. . . . . . . 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 17625	. . . . . . 7 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ ∪ (𝐹 “ 𝑡) ∈ 𝐶) → (𝐹‘∪ (𝐹 “ 𝑡)) = ∪ (𝐹 “ 𝑡))
36	1, 34, 35	syl2anc 584	. . . . . 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 2770	. . . . 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 3131	. . 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 1540   ∈ wcel 2108  ∀wral 3051  Vcvv 3459   ⊆ wss 3926  𝒫 cpw 4575  ∪ cuni 4883  ran crn 5655   “ cima 5657   Fn wfn 6526  ‘cfv 6531  Moorecmre 17594  mrClscmrc 17595  Dirsetcdrs 18305  toInccipo 18537

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729  ax-cnex 11185  ax-resscn 11186  ax-1cn 11187  ax-icn 11188  ax-addcl 11189  ax-addrcl 11190  ax-mulcl 11191  ax-mulrcl 11192  ax-mulcom 11193  ax-addass 11194  ax-mulass 11195  ax-distr 11196  ax-i2m1 11197  ax-1ne0 11198  ax-1rid 11199  ax-rnegex 11200  ax-rrecex 11201  ax-cnre 11202  ax-pre-lttri 11203  ax-pre-lttrn 11204  ax-pre-ltadd 11205  ax-pre-mulgt0 11206

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 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-int 4923  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7362  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7862  df-1st 7988  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-er 8719  df-en 8960  df-dom 8961  df-sdom 8962  df-fin 8963  df-pnf 11271  df-mnf 11272  df-xr 11273  df-ltxr 11274  df-le 11275  df-sub 11468  df-neg 11469  df-nn 12241  df-2 12303  df-3 12304  df-4 12305  df-5 12306  df-6 12307  df-7 12308  df-8 12309  df-9 12310  df-n0 12502  df-z 12589  df-dec 12709  df-uz 12853  df-fz 13525  df-struct 17166  df-slot 17201  df-ndx 17213  df-base 17229  df-tset 17290  df-ple 17291  df-ocomp 17292  df-mre 17598  df-mrc 17599  df-proset 18306  df-drs 18307  df-poset 18325  df-ipo 18538

This theorem is referenced by:  acsdrscl  18556  acsficl  18557  isacs5  18558  isacs4  18559  isacs3  18560