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Theorem acsmap2d 18404
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 18403 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 17480, 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 (𝜑𝐴 ∈ (ACS‘𝑋))
acsmap2d.2 𝑁 = (mrCls‘𝐴)
acsmap2d.3 𝐼 = (mrInd‘𝐴)
acsmap2d.4 (𝜑𝑆𝐼)
acsmap2d.5 (𝜑𝑇𝑋)
acsmap2d.6 (𝜑 → (𝑁𝑆) = (𝑁𝑇))
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
StepHypRef Expression
1 acsmap2d.1 . . 3 (𝜑𝐴 ∈ (ACS‘𝑋))
2 acsmap2d.2 . . 3 𝑁 = (mrCls‘𝐴)
3 acsmap2d.3 . . . 4 𝐼 = (mrInd‘𝐴)
41acsmred 17496 . . . 4 (𝜑𝐴 ∈ (Moore‘𝑋))
5 acsmap2d.4 . . . 4 (𝜑𝑆𝐼)
63, 4, 5mrissd 17476 . . 3 (𝜑𝑆𝑋)
7 acsmap2d.5 . . . . 5 (𝜑𝑇𝑋)
84, 2, 7mrcssidd 17465 . . . 4 (𝜑𝑇 ⊆ (𝑁𝑇))
9 acsmap2d.6 . . . 4 (𝜑 → (𝑁𝑆) = (𝑁𝑇))
108, 9sseqtrrd 3983 . . 3 (𝜑𝑇 ⊆ (𝑁𝑆))
111, 2, 6, 10acsmapd 18403 . 2 (𝜑 → ∃𝑓(𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑇 ⊆ (𝑁 ran 𝑓)))
12 simprl 769 . . . . 5 ((𝜑 ∧ (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑇 ⊆ (𝑁 ran 𝑓))) → 𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin))
134adantr 481 . . . . . 6 ((𝜑 ∧ (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑇 ⊆ (𝑁 ran 𝑓))) → 𝐴 ∈ (Moore‘𝑋))
145adantr 481 . . . . . . . . 9 ((𝜑 ∧ (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑇 ⊆ (𝑁 ran 𝑓))) → 𝑆𝐼)
153, 13, 14mrissd 17476 . . . . . . . 8 ((𝜑 ∧ (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑇 ⊆ (𝑁 ran 𝑓))) → 𝑆𝑋)
1613, 2, 15mrcssidd 17465 . . . . . . 7 ((𝜑 ∧ (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑇 ⊆ (𝑁 ran 𝑓))) → 𝑆 ⊆ (𝑁𝑆))
179adantr 481 . . . . . . . 8 ((𝜑 ∧ (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑇 ⊆ (𝑁 ran 𝑓))) → (𝑁𝑆) = (𝑁𝑇))
18 simprr 771 . . . . . . . . . 10 ((𝜑 ∧ (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑇 ⊆ (𝑁 ran 𝑓))) → 𝑇 ⊆ (𝑁 ran 𝑓))
1913, 2mrcssvd 17463 . . . . . . . . . 10 ((𝜑 ∧ (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑇 ⊆ (𝑁 ran 𝑓))) → (𝑁 ran 𝑓) ⊆ 𝑋)
2013, 2, 18, 19mrcssd 17464 . . . . . . . . 9 ((𝜑 ∧ (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑇 ⊆ (𝑁 ran 𝑓))) → (𝑁𝑇) ⊆ (𝑁‘(𝑁 ran 𝑓)))
21 frn 6672 . . . . . . . . . . . . . 14 (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) → ran 𝑓 ⊆ (𝒫 𝑆 ∩ Fin))
2221unissd 4873 . . . . . . . . . . . . 13 (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) → ran 𝑓 (𝒫 𝑆 ∩ Fin))
23 unifpw 9257 . . . . . . . . . . . . 13 (𝒫 𝑆 ∩ Fin) = 𝑆
2422, 23sseqtrdi 3992 . . . . . . . . . . . 12 (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) → ran 𝑓𝑆)
2524ad2antrl 726 . . . . . . . . . . 11 ((𝜑 ∧ (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑇 ⊆ (𝑁 ran 𝑓))) → ran 𝑓𝑆)
2625, 15sstrd 3952 . . . . . . . . . 10 ((𝜑 ∧ (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑇 ⊆ (𝑁 ran 𝑓))) → ran 𝑓𝑋)
2713, 2, 26mrcidmd 17466 . . . . . . . . 9 ((𝜑 ∧ (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑇 ⊆ (𝑁 ran 𝑓))) → (𝑁‘(𝑁 ran 𝑓)) = (𝑁 ran 𝑓))
2820, 27sseqtrd 3982 . . . . . . . 8 ((𝜑 ∧ (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑇 ⊆ (𝑁 ran 𝑓))) → (𝑁𝑇) ⊆ (𝑁 ran 𝑓))
2917, 28eqsstrd 3980 . . . . . . 7 ((𝜑 ∧ (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑇 ⊆ (𝑁 ran 𝑓))) → (𝑁𝑆) ⊆ (𝑁 ran 𝑓))
3016, 29sstrd 3952 . . . . . 6 ((𝜑 ∧ (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑇 ⊆ (𝑁 ran 𝑓))) → 𝑆 ⊆ (𝑁 ran 𝑓))
3113, 2, 3, 30, 25, 14mrissmrcd 17480 . . . . 5 ((𝜑 ∧ (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑇 ⊆ (𝑁 ran 𝑓))) → 𝑆 = ran 𝑓)
3212, 31jca 512 . . . 4 ((𝜑 ∧ (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑇 ⊆ (𝑁 ran 𝑓))) → (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑆 = ran 𝑓))
3332ex 413 . . 3 (𝜑 → ((𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑇 ⊆ (𝑁 ran 𝑓)) → (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑆 = ran 𝑓)))
3433eximdv 1920 . 2 (𝜑 → (∃𝑓(𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑇 ⊆ (𝑁 ran 𝑓)) → ∃𝑓(𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑆 = ran 𝑓)))
3511, 34mpd 15 1 (𝜑 → ∃𝑓(𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑆 = ran 𝑓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wex 1781  wcel 2106  cin 3907  wss 3908  𝒫 cpw 4558   cuni 4863  ran crn 5632  wf 6489  cfv 6493  Fincfn 8841  Moorecmre 17422  mrClscmrc 17423  mrIndcmri 17424  ACScacs 17425
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708  ax-rep 5240  ax-sep 5254  ax-nul 5261  ax-pow 5318  ax-pr 5382  ax-un 7664  ax-reg 9486  ax-inf2 9535  ax-ac2 10357  ax-cnex 11065  ax-resscn 11066  ax-1cn 11067  ax-icn 11068  ax-addcl 11069  ax-addrcl 11070  ax-mulcl 11071  ax-mulrcl 11072  ax-mulcom 11073  ax-addass 11074  ax-mulass 11075  ax-distr 11076  ax-i2m1 11077  ax-1ne0 11078  ax-1rid 11079  ax-rnegex 11080  ax-rrecex 11081  ax-cnre 11082  ax-pre-lttri 11083  ax-pre-lttrn 11084  ax-pre-ltadd 11085  ax-pre-mulgt0 11086
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3351  df-reu 3352  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4281  df-if 4485  df-pw 4560  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-int 4906  df-iun 4954  df-iin 4955  df-br 5104  df-opab 5166  df-mpt 5187  df-tr 5221  df-id 5529  df-eprel 5535  df-po 5543  df-so 5544  df-fr 5586  df-se 5587  df-we 5588  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 6251  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6445  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-isom 6502  df-riota 7307  df-ov 7354  df-oprab 7355  df-mpo 7356  df-om 7795  df-1st 7913  df-2nd 7914  df-frecs 8204  df-wrecs 8235  df-recs 8309  df-rdg 8348  df-1o 8404  df-er 8606  df-en 8842  df-dom 8843  df-sdom 8844  df-fin 8845  df-r1 9658  df-rank 9659  df-card 9833  df-ac 10010  df-pnf 11149  df-mnf 11150  df-xr 11151  df-ltxr 11152  df-le 11153  df-sub 11345  df-neg 11346  df-nn 12112  df-2 12174  df-3 12175  df-4 12176  df-5 12177  df-6 12178  df-7 12179  df-8 12180  df-9 12181  df-n0 12372  df-z 12458  df-dec 12577  df-uz 12722  df-fz 13379  df-struct 16979  df-slot 17014  df-ndx 17026  df-base 17044  df-tset 17112  df-ple 17113  df-ocomp 17114  df-mre 17426  df-mrc 17427  df-mri 17428  df-acs 17429  df-proset 18144  df-drs 18145  df-poset 18162  df-ipo 18377
This theorem is referenced by:  acsinfd  18405  acsdomd  18406
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