Theorem acsmap2d 18600

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 18599 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 17683, ∪ 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 17699	. . . 4 ⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋))
5		acsmap2d.4	. . . 4 ⊢ (𝜑 → 𝑆 ∈ 𝐼)
6	3, 4, 5	mrissd 17679	. . 3 ⊢ (𝜑 → 𝑆 ⊆ 𝑋)
7		acsmap2d.5	. . . . 5 ⊢ (𝜑 → 𝑇 ⊆ 𝑋)
8	4, 2, 7	mrcssidd 17668	. . . 4 ⊢ (𝜑 → 𝑇 ⊆ (𝑁‘𝑇))
9		acsmap2d.6	. . . 4 ⊢ (𝜑 → (𝑁‘𝑆) = (𝑁‘𝑇))
10	8, 9	sseqtrrd 4021	. . 3 ⊢ (𝜑 → 𝑇 ⊆ (𝑁‘𝑆))
11	1, 2, 6, 10	acsmapd 18599	. 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 17679	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑇 ⊆ (𝑁‘∪ ran 𝑓))) → 𝑆 ⊆ 𝑋)
16	13, 2, 15	mrcssidd 17668	. . . . . . 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 17666	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑇 ⊆ (𝑁‘∪ ran 𝑓))) → (𝑁‘∪ ran 𝑓) ⊆ 𝑋)
20	13, 2, 18, 19	mrcssd 17667	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑇 ⊆ (𝑁‘∪ ran 𝑓))) → (𝑁‘𝑇) ⊆ (𝑁‘(𝑁‘∪ ran 𝑓)))
21		frn 6743	. . . . . . . . . . . . . 14 ⊢ (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) → ran 𝑓 ⊆ (𝒫 𝑆 ∩ Fin))
22	21	unissd 4917	. . . . . . . . . . . . 13 ⊢ (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) → ∪ ran 𝑓 ⊆ ∪ (𝒫 𝑆 ∩ Fin))
23		unifpw 9395	. . . . . . . . . . . . 13 ⊢ ∪ (𝒫 𝑆 ∩ Fin) = 𝑆
24	22, 23	sseqtrdi 4024	. . . . . . . . . . . 12 ⊢ (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) → ∪ ran 𝑓 ⊆ 𝑆)
25	24	ad2antrl 728	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑇 ⊆ (𝑁‘∪ ran 𝑓))) → ∪ ran 𝑓 ⊆ 𝑆)
26	25, 15	sstrd 3994	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑇 ⊆ (𝑁‘∪ ran 𝑓))) → ∪ ran 𝑓 ⊆ 𝑋)
27	13, 2, 26	mrcidmd 17669	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑇 ⊆ (𝑁‘∪ ran 𝑓))) → (𝑁‘(𝑁‘∪ ran 𝑓)) = (𝑁‘∪ ran 𝑓))
28	20, 27	sseqtrd 4020	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑇 ⊆ (𝑁‘∪ ran 𝑓))) → (𝑁‘𝑇) ⊆ (𝑁‘∪ ran 𝑓))
29	17, 28	eqsstrd 4018	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑇 ⊆ (𝑁‘∪ ran 𝑓))) → (𝑁‘𝑆) ⊆ (𝑁‘∪ ran 𝑓))
30	16, 29	sstrd 3994	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑇 ⊆ (𝑁‘∪ ran 𝑓))) → 𝑆 ⊆ (𝑁‘∪ ran 𝑓))
31	13, 2, 3, 30, 25, 14	mrissmrcd 17683	. . . . 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 1917	. 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 1540  ∃wex 1779   ∈ wcel 2108   ∩ cin 3950   ⊆ wss 3951  𝒫 cpw 4600  ∪ cuni 4907  ran crn 5686  ⟶wf 6557  ‘cfv 6561  Fincfn 8985  Moorecmre 17625  mrClscmrc 17626  mrIndcmri 17627  ACScacs 17628

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 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-reg 9632  ax-inf2 9681  ax-ac2 10503  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-r1 9804  df-rank 9805  df-card 9979  df-ac 10156  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-7 12334  df-8 12335  df-9 12336  df-n0 12527  df-z 12614  df-dec 12734  df-uz 12879  df-fz 13548  df-struct 17184  df-slot 17219  df-ndx 17231  df-base 17248  df-tset 17316  df-ple 17317  df-ocomp 17318  df-mre 17629  df-mrc 17630  df-mri 17631  df-acs 17632  df-proset 18340  df-drs 18341  df-poset 18359  df-ipo 18573

This theorem is referenced by:  acsinfd  18601  acsdomd  18602