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Theorem acsmap2d 17002
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 17001 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 16123, 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 16140 . . . 4 (𝜑𝐴 ∈ (Moore‘𝑋))
5 acsmap2d.4 . . . 4 (𝜑𝑆𝐼)
63, 4, 5mrissd 16119 . . 3 (𝜑𝑆𝑋)
7 acsmap2d.5 . . . . 5 (𝜑𝑇𝑋)
84, 2, 7mrcssidd 16108 . . . 4 (𝜑𝑇 ⊆ (𝑁𝑇))
9 acsmap2d.6 . . . 4 (𝜑 → (𝑁𝑆) = (𝑁𝑇))
108, 9sseqtr4d 3605 . . 3 (𝜑𝑇 ⊆ (𝑁𝑆))
111, 2, 6, 10acsmapd 17001 . 2 (𝜑 → ∃𝑓(𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑇 ⊆ (𝑁 ran 𝑓)))
12 simprl 790 . . . . 5 ((𝜑 ∧ (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑇 ⊆ (𝑁 ran 𝑓))) → 𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin))
134adantr 480 . . . . . 6 ((𝜑 ∧ (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑇 ⊆ (𝑁 ran 𝑓))) → 𝐴 ∈ (Moore‘𝑋))
145adantr 480 . . . . . . . . 9 ((𝜑 ∧ (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑇 ⊆ (𝑁 ran 𝑓))) → 𝑆𝐼)
153, 13, 14mrissd 16119 . . . . . . . 8 ((𝜑 ∧ (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑇 ⊆ (𝑁 ran 𝑓))) → 𝑆𝑋)
1613, 2, 15mrcssidd 16108 . . . . . . 7 ((𝜑 ∧ (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑇 ⊆ (𝑁 ran 𝑓))) → 𝑆 ⊆ (𝑁𝑆))
179adantr 480 . . . . . . . 8 ((𝜑 ∧ (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑇 ⊆ (𝑁 ran 𝑓))) → (𝑁𝑆) = (𝑁𝑇))
18 simprr 792 . . . . . . . . . 10 ((𝜑 ∧ (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑇 ⊆ (𝑁 ran 𝑓))) → 𝑇 ⊆ (𝑁 ran 𝑓))
1913, 2mrcssvd 16106 . . . . . . . . . 10 ((𝜑 ∧ (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑇 ⊆ (𝑁 ran 𝑓))) → (𝑁 ran 𝑓) ⊆ 𝑋)
2013, 2, 18, 19mrcssd 16107 . . . . . . . . 9 ((𝜑 ∧ (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑇 ⊆ (𝑁 ran 𝑓))) → (𝑁𝑇) ⊆ (𝑁‘(𝑁 ran 𝑓)))
21 frn 5966 . . . . . . . . . . . . . 14 (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) → ran 𝑓 ⊆ (𝒫 𝑆 ∩ Fin))
2221unissd 4398 . . . . . . . . . . . . 13 (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) → ran 𝑓 (𝒫 𝑆 ∩ Fin))
23 unifpw 8152 . . . . . . . . . . . . 13 (𝒫 𝑆 ∩ Fin) = 𝑆
2422, 23syl6sseq 3614 . . . . . . . . . . . 12 (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) → ran 𝑓𝑆)
2524ad2antrl 760 . . . . . . . . . . 11 ((𝜑 ∧ (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑇 ⊆ (𝑁 ran 𝑓))) → ran 𝑓𝑆)
2625, 15sstrd 3578 . . . . . . . . . 10 ((𝜑 ∧ (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑇 ⊆ (𝑁 ran 𝑓))) → ran 𝑓𝑋)
2713, 2, 26mrcidmd 16109 . . . . . . . . 9 ((𝜑 ∧ (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑇 ⊆ (𝑁 ran 𝑓))) → (𝑁‘(𝑁 ran 𝑓)) = (𝑁 ran 𝑓))
2820, 27sseqtrd 3604 . . . . . . . 8 ((𝜑 ∧ (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑇 ⊆ (𝑁 ran 𝑓))) → (𝑁𝑇) ⊆ (𝑁 ran 𝑓))
2917, 28eqsstrd 3602 . . . . . . 7 ((𝜑 ∧ (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑇 ⊆ (𝑁 ran 𝑓))) → (𝑁𝑆) ⊆ (𝑁 ran 𝑓))
3016, 29sstrd 3578 . . . . . 6 ((𝜑 ∧ (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑇 ⊆ (𝑁 ran 𝑓))) → 𝑆 ⊆ (𝑁 ran 𝑓))
3113, 2, 3, 30, 25, 14mrissmrcd 16123 . . . . 5 ((𝜑 ∧ (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑇 ⊆ (𝑁 ran 𝑓))) → 𝑆 = ran 𝑓)
3212, 31jca 553 . . . 4 ((𝜑 ∧ (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑇 ⊆ (𝑁 ran 𝑓))) → (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑆 = ran 𝑓))
3332ex 449 . . 3 (𝜑 → ((𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑇 ⊆ (𝑁 ran 𝑓)) → (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑆 = ran 𝑓)))
3433eximdv 1833 . 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 383   = wceq 1475  wex 1695  wcel 1977  cin 3539  wss 3540  𝒫 cpw 4108   cuni 4372  ran crn 5039  wf 5800  cfv 5804  Fincfn 7841  Moorecmre 16065  mrClscmrc 16066  mrIndcmri 16067  ACScacs 16068
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-reg 8380  ax-inf2 8421  ax-ac2 9168  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-r1 8510  df-rank 8511  df-card 8648  df-ac 8822  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-dec 11370  df-uz 11564  df-fz 12198  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-tset 15787  df-ple 15788  df-ocomp 15790  df-mre 16069  df-mrc 16070  df-mri 16071  df-acs 16072  df-preset 16751  df-drs 16752  df-poset 16769  df-ipo 16975
This theorem is referenced by:  acsinfd  17003  acsdomd  17004
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