Theorem acsmap2d 18512

Description: In an algebraic closure system, if 𝑆 and 𝑇 have the same closure and 𝑆 is independent, then there is a map 𝑓 from 𝑇 into the set of finite subsets of 𝑆 such that 𝑆 equals the union of ran 𝑓. This is proven by taking the map 𝑓 from acsmapd 18511 and observing that, since 𝑆 and 𝑇 have the same closure, the closure of ∪ ran 𝑓 must contain 𝑆. Since 𝑆 is independent, by mrissmrcd 17597, ∪ ran 𝑓 must equal 𝑆. See Section II.5 in [Cohn] p. 81 to 82. (Contributed by David Moews, 1-May-2017.)

Hypotheses

Ref	Expression
acsmap2d.1	⊢ (𝜑 → 𝐴 ∈ (ACS‘𝑋))
acsmap2d.2	⊢ 𝑁 = (mrCls‘𝐴)
acsmap2d.3	⊢ 𝐼 = (mrInd‘𝐴)
acsmap2d.4	⊢ (𝜑 → 𝑆 ∈ 𝐼)
acsmap2d.5	⊢ (𝜑 → 𝑇 ⊆ 𝑋)
acsmap2d.6	⊢ (𝜑 → (𝑁‘𝑆) = (𝑁‘𝑇))

Assertion

Ref	Expression
acsmap2d	⊢ (𝜑 → ∃𝑓(𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑆 = ∪ ran 𝑓))

Distinct variable groups:   𝑆,𝑓   𝑇,𝑓   𝜑,𝑓   𝑓,𝑁

Allowed substitution hints:   𝐴(𝑓)   𝐼(𝑓)   𝑋(𝑓)

Proof of Theorem acsmap2d

Step	Hyp	Ref	Expression
1		acsmap2d.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ (ACS‘𝑋))
2		acsmap2d.2	. . 3 ⊢ 𝑁 = (mrCls‘𝐴)
3		acsmap2d.3	. . . 4 ⊢ 𝐼 = (mrInd‘𝐴)
4	1	acsmred 17613	. . . 4 ⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋))
5		acsmap2d.4	. . . 4 ⊢ (𝜑 → 𝑆 ∈ 𝐼)
6	3, 4, 5	mrissd 17593	. . 3 ⊢ (𝜑 → 𝑆 ⊆ 𝑋)
7		acsmap2d.5	. . . . 5 ⊢ (𝜑 → 𝑇 ⊆ 𝑋)
8	4, 2, 7	mrcssidd 17582	. . . 4 ⊢ (𝜑 → 𝑇 ⊆ (𝑁‘𝑇))
9		acsmap2d.6	. . . 4 ⊢ (𝜑 → (𝑁‘𝑆) = (𝑁‘𝑇))
10	8, 9	sseqtrrd 3960	. . 3 ⊢ (𝜑 → 𝑇 ⊆ (𝑁‘𝑆))
11	1, 2, 6, 10	acsmapd 18511	. 2 ⊢ (𝜑 → ∃𝑓(𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑇 ⊆ (𝑁‘∪ ran 𝑓)))
12		simprl 771	. . . . 5 ⊢ ((𝜑 ∧ (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑇 ⊆ (𝑁‘∪ ran 𝑓))) → 𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin))
13	4	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑇 ⊆ (𝑁‘∪ ran 𝑓))) → 𝐴 ∈ (Moore‘𝑋))
14	5	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑇 ⊆ (𝑁‘∪ ran 𝑓))) → 𝑆 ∈ 𝐼)
15	3, 13, 14	mrissd 17593	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑇 ⊆ (𝑁‘∪ ran 𝑓))) → 𝑆 ⊆ 𝑋)
16	13, 2, 15	mrcssidd 17582	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑇 ⊆ (𝑁‘∪ ran 𝑓))) → 𝑆 ⊆ (𝑁‘𝑆))
17	9	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑇 ⊆ (𝑁‘∪ ran 𝑓))) → (𝑁‘𝑆) = (𝑁‘𝑇))
18		simprr 773	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑇 ⊆ (𝑁‘∪ ran 𝑓))) → 𝑇 ⊆ (𝑁‘∪ ran 𝑓))
19	13, 2	mrcssvd 17580	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑇 ⊆ (𝑁‘∪ ran 𝑓))) → (𝑁‘∪ ran 𝑓) ⊆ 𝑋)
20	13, 2, 18, 19	mrcssd 17581	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑇 ⊆ (𝑁‘∪ ran 𝑓))) → (𝑁‘𝑇) ⊆ (𝑁‘(𝑁‘∪ ran 𝑓)))
21		frn 6669	. . . . . . . . . . . . . 14 ⊢ (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) → ran 𝑓 ⊆ (𝒫 𝑆 ∩ Fin))
22	21	unissd 4861	. . . . . . . . . . . . 13 ⊢ (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) → ∪ ran 𝑓 ⊆ ∪ (𝒫 𝑆 ∩ Fin))
23		unifpw 9258	. . . . . . . . . . . . 13 ⊢ ∪ (𝒫 𝑆 ∩ Fin) = 𝑆
24	22, 23	sseqtrdi 3963	. . . . . . . . . . . 12 ⊢ (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) → ∪ ran 𝑓 ⊆ 𝑆)
25	24	ad2antrl 729	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑇 ⊆ (𝑁‘∪ ran 𝑓))) → ∪ ran 𝑓 ⊆ 𝑆)
26	25, 15	sstrd 3933	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑇 ⊆ (𝑁‘∪ ran 𝑓))) → ∪ ran 𝑓 ⊆ 𝑋)
27	13, 2, 26	mrcidmd 17583	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑇 ⊆ (𝑁‘∪ ran 𝑓))) → (𝑁‘(𝑁‘∪ ran 𝑓)) = (𝑁‘∪ ran 𝑓))
28	20, 27	sseqtrd 3959	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑇 ⊆ (𝑁‘∪ ran 𝑓))) → (𝑁‘𝑇) ⊆ (𝑁‘∪ ran 𝑓))
29	17, 28	eqsstrd 3957	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑇 ⊆ (𝑁‘∪ ran 𝑓))) → (𝑁‘𝑆) ⊆ (𝑁‘∪ ran 𝑓))
30	16, 29	sstrd 3933	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑇 ⊆ (𝑁‘∪ ran 𝑓))) → 𝑆 ⊆ (𝑁‘∪ ran 𝑓))
31	13, 2, 3, 30, 25, 14	mrissmrcd 17597	. . . . 5 ⊢ ((𝜑 ∧ (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑇 ⊆ (𝑁‘∪ ran 𝑓))) → 𝑆 = ∪ ran 𝑓)
32	12, 31	jca 511	. . . 4 ⊢ ((𝜑 ∧ (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑇 ⊆ (𝑁‘∪ ran 𝑓))) → (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑆 = ∪ ran 𝑓))
33	32	ex 412	. . 3 ⊢ (𝜑 → ((𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑇 ⊆ (𝑁‘∪ ran 𝑓)) → (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑆 = ∪ ran 𝑓)))
34	33	eximdv 1919	. 2 ⊢ (𝜑 → (∃𝑓(𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑇 ⊆ (𝑁‘∪ ran 𝑓)) → ∃𝑓(𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑆 = ∪ ran 𝑓)))
35	11, 34	mpd 15	1 ⊢ (𝜑 → ∃𝑓(𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑆 = ∪ ran 𝑓))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542  ∃wex 1781   ∈ wcel 2114   ∩ cin 3889   ⊆ wss 3890  𝒫 cpw 4542  ∪ cuni 4851  ran crn 5625  ⟶wf 6488  ‘cfv 6492  Fincfn 8886  Moorecmre 17535  mrClscmrc 17536  mrIndcmri 17537  ACScacs 17538

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 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682  ax-reg 9500  ax-inf2 9553  ax-ac2 10376  ax-cnex 11085  ax-resscn 11086  ax-1cn 11087  ax-icn 11088  ax-addcl 11089  ax-addrcl 11090  ax-mulcl 11091  ax-mulrcl 11092  ax-mulcom 11093  ax-addass 11094  ax-mulass 11095  ax-distr 11096  ax-i2m1 11097  ax-1ne0 11098  ax-1rid 11099  ax-rnegex 11100  ax-rrecex 11101  ax-cnre 11102  ax-pre-lttri 11103  ax-pre-lttrn 11104  ax-pre-ltadd 11105  ax-pre-mulgt0 11106

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 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-iin 4937  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-se 5578  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-isom 6501  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-1st 7935  df-2nd 7936  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-rdg 8342  df-1o 8398  df-er 8636  df-en 8887  df-dom 8888  df-sdom 8889  df-fin 8890  df-r1 9679  df-rank 9680  df-card 9854  df-ac 10029  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-nn 12166  df-2 12235  df-3 12236  df-4 12237  df-5 12238  df-6 12239  df-7 12240  df-8 12241  df-9 12242  df-n0 12429  df-z 12516  df-dec 12636  df-uz 12780  df-fz 13453  df-struct 17108  df-slot 17143  df-ndx 17155  df-base 17171  df-tset 17230  df-ple 17231  df-ocomp 17232  df-mre 17539  df-mrc 17540  df-mri 17541  df-acs 17542  df-proset 18251  df-drs 18252  df-poset 18270  df-ipo 18485

This theorem is referenced by:  acsinfd  18513  acsdomd  18514