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Theorem acsmap2d 18613
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 18612 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 17685, 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 17701 . . . 4 (𝜑𝐴 ∈ (Moore‘𝑋))
5 acsmap2d.4 . . . 4 (𝜑𝑆𝐼)
63, 4, 5mrissd 17681 . . 3 (𝜑𝑆𝑋)
7 acsmap2d.5 . . . . 5 (𝜑𝑇𝑋)
84, 2, 7mrcssidd 17670 . . . 4 (𝜑𝑇 ⊆ (𝑁𝑇))
9 acsmap2d.6 . . . 4 (𝜑 → (𝑁𝑆) = (𝑁𝑇))
108, 9sseqtrrd 4037 . . 3 (𝜑𝑇 ⊆ (𝑁𝑆))
111, 2, 6, 10acsmapd 18612 . 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))
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 17681 . . . . . . . 8 ((𝜑 ∧ (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑇 ⊆ (𝑁 ran 𝑓))) → 𝑆𝑋)
1613, 2, 15mrcssidd 17670 . . . . . . 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 773 . . . . . . . . . 10 ((𝜑 ∧ (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑇 ⊆ (𝑁 ran 𝑓))) → 𝑇 ⊆ (𝑁 ran 𝑓))
1913, 2mrcssvd 17668 . . . . . . . . . 10 ((𝜑 ∧ (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑇 ⊆ (𝑁 ran 𝑓))) → (𝑁 ran 𝑓) ⊆ 𝑋)
2013, 2, 18, 19mrcssd 17669 . . . . . . . . 9 ((𝜑 ∧ (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑇 ⊆ (𝑁 ran 𝑓))) → (𝑁𝑇) ⊆ (𝑁‘(𝑁 ran 𝑓)))
21 frn 6744 . . . . . . . . . . . . . 14 (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) → ran 𝑓 ⊆ (𝒫 𝑆 ∩ Fin))
2221unissd 4922 . . . . . . . . . . . . 13 (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) → ran 𝑓 (𝒫 𝑆 ∩ Fin))
23 unifpw 9393 . . . . . . . . . . . . 13 (𝒫 𝑆 ∩ Fin) = 𝑆
2422, 23sseqtrdi 4046 . . . . . . . . . . . 12 (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) → ran 𝑓𝑆)
2524ad2antrl 728 . . . . . . . . . . 11 ((𝜑 ∧ (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑇 ⊆ (𝑁 ran 𝑓))) → ran 𝑓𝑆)
2625, 15sstrd 4006 . . . . . . . . . 10 ((𝜑 ∧ (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑇 ⊆ (𝑁 ran 𝑓))) → ran 𝑓𝑋)
2713, 2, 26mrcidmd 17671 . . . . . . . . 9 ((𝜑 ∧ (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑇 ⊆ (𝑁 ran 𝑓))) → (𝑁‘(𝑁 ran 𝑓)) = (𝑁 ran 𝑓))
2820, 27sseqtrd 4036 . . . . . . . 8 ((𝜑 ∧ (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑇 ⊆ (𝑁 ran 𝑓))) → (𝑁𝑇) ⊆ (𝑁 ran 𝑓))
2917, 28eqsstrd 4034 . . . . . . 7 ((𝜑 ∧ (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑇 ⊆ (𝑁 ran 𝑓))) → (𝑁𝑆) ⊆ (𝑁 ran 𝑓))
3016, 29sstrd 4006 . . . . . 6 ((𝜑 ∧ (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑇 ⊆ (𝑁 ran 𝑓))) → 𝑆 ⊆ (𝑁 ran 𝑓))
3113, 2, 3, 30, 25, 14mrissmrcd 17685 . . . . 5 ((𝜑 ∧ (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑇 ⊆ (𝑁 ran 𝑓))) → 𝑆 = ran 𝑓)
3212, 31jca 511 . . . 4 ((𝜑 ∧ (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑇 ⊆ (𝑁 ran 𝑓))) → (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑆 = ran 𝑓))
3332ex 412 . . 3 (𝜑 → ((𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑇 ⊆ (𝑁 ran 𝑓)) → (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑆 = ran 𝑓)))
3433eximdv 1915 . 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 395   = wceq 1537  wex 1776  wcel 2106  cin 3962  wss 3963  𝒫 cpw 4605   cuni 4912  ran crn 5690  wf 6559  cfv 6563  Fincfn 8984  Moorecmre 17627  mrClscmrc 17628  mrIndcmri 17629  ACScacs 17630
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-reg 9630  ax-inf2 9679  ax-ac2 10501  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-r1 9802  df-rank 9803  df-card 9977  df-ac 10154  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12525  df-z 12612  df-dec 12732  df-uz 12877  df-fz 13545  df-struct 17181  df-slot 17216  df-ndx 17228  df-base 17246  df-tset 17317  df-ple 17318  df-ocomp 17319  df-mre 17631  df-mrc 17632  df-mri 17633  df-acs 17634  df-proset 18352  df-drs 18353  df-poset 18371  df-ipo 18586
This theorem is referenced by:  acsinfd  18614  acsdomd  18615
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