MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  acsmap2d Structured version   Visualization version   GIF version
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 βˆͺ ran 𝑓 must contain 𝑆. Since 𝑆 is independent, by mrissmrcd 17588, βˆͺ 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 (πœ‘ β†’ 𝐴 ∈ (ACSβ€˜π‘‹))
acsmap2d.2 𝑁 = (mrClsβ€˜π΄)
acsmap2d.3 𝐼 = (mrIndβ€˜π΄)
acsmap2d.4 (πœ‘ β†’ 𝑆 ∈ 𝐼)
acsmap2d.5 (πœ‘ β†’ 𝑇 βŠ† 𝑋)
acsmap2d.6 (πœ‘ β†’ (π‘β€˜π‘†) = (π‘β€˜π‘‡))
Assertion
Ref Expression
acsmap2d (πœ‘ β†’ βˆƒπ‘“(𝑓:π‘‡βŸΆ(𝒫 𝑆 ∩ Fin) ∧ 𝑆 = βˆͺ ran 𝑓))
Distinct variable groups:   𝑆,𝑓   𝑇,𝑓   πœ‘,𝑓   𝑓,𝑁
Allowed substitution hints:   𝐴(𝑓)   𝐼(𝑓)   𝑋(𝑓)
Proof of Theorem acsmap2d
StepHypRef Expression
1 acsmap2d.1 . . 3 (πœ‘ β†’ 𝐴 ∈ (ACSβ€˜π‘‹))
2 acsmap2d.2 . . 3 𝑁 = (mrClsβ€˜π΄)
3 acsmap2d.3 . . . 4 𝐼 = (mrIndβ€˜π΄)
41acsmred 17604 . . . 4 (πœ‘ β†’ 𝐴 ∈ (Mooreβ€˜π‘‹))
5 acsmap2d.4 . . . 4 (πœ‘ β†’ 𝑆 ∈ 𝐼)
63, 4, 5mrissd 17584 . . 3 (πœ‘ β†’ 𝑆 βŠ† 𝑋)
7 acsmap2d.5 . . . . 5 (πœ‘ β†’ 𝑇 βŠ† 𝑋)
84, 2, 7mrcssidd 17573 . . . 4 (πœ‘ β†’ 𝑇 βŠ† (π‘β€˜π‘‡))
9 acsmap2d.6 . . . 4 (πœ‘ β†’ (π‘β€˜π‘†) = (π‘β€˜π‘‡))
108, 9sseqtrrd 4023 . . 3 (πœ‘ β†’ 𝑇 βŠ† (π‘β€˜π‘†))
111, 2, 6, 10acsmapd 18511 . 2 (πœ‘ β†’ βˆƒπ‘“(𝑓:π‘‡βŸΆ(𝒫 𝑆 ∩ Fin) ∧ 𝑇 βŠ† (π‘β€˜βˆͺ ran 𝑓)))
12 simprl 769 . . . . 5 ((πœ‘ ∧ (𝑓:π‘‡βŸΆ(𝒫 𝑆 ∩ Fin) ∧ 𝑇 βŠ† (π‘β€˜βˆͺ ran 𝑓))) β†’ 𝑓:π‘‡βŸΆ(𝒫 𝑆 ∩ Fin))
134adantr 481 . . . . . 6 ((πœ‘ ∧ (𝑓:π‘‡βŸΆ(𝒫 𝑆 ∩ Fin) ∧ 𝑇 βŠ† (π‘β€˜βˆͺ ran 𝑓))) β†’ 𝐴 ∈ (Mooreβ€˜π‘‹))
145adantr 481 . . . . . . . . 9 ((πœ‘ ∧ (𝑓:π‘‡βŸΆ(𝒫 𝑆 ∩ Fin) ∧ 𝑇 βŠ† (π‘β€˜βˆͺ ran 𝑓))) β†’ 𝑆 ∈ 𝐼)
153, 13, 14mrissd 17584 . . . . . . . 8 ((πœ‘ ∧ (𝑓:π‘‡βŸΆ(𝒫 𝑆 ∩ Fin) ∧ 𝑇 βŠ† (π‘β€˜βˆͺ ran 𝑓))) β†’ 𝑆 βŠ† 𝑋)
1613, 2, 15mrcssidd 17573 . . . . . . 7 ((πœ‘ ∧ (𝑓:π‘‡βŸΆ(𝒫 𝑆 ∩ Fin) ∧ 𝑇 βŠ† (π‘β€˜βˆͺ ran 𝑓))) β†’ 𝑆 βŠ† (π‘β€˜π‘†))
179adantr 481 . . . . . . . 8 ((πœ‘ ∧ (𝑓:π‘‡βŸΆ(𝒫 𝑆 ∩ Fin) ∧ 𝑇 βŠ† (π‘β€˜βˆͺ ran 𝑓))) β†’ (π‘β€˜π‘†) = (π‘β€˜π‘‡))
18 simprr 771 . . . . . . . . . 10 ((πœ‘ ∧ (𝑓:π‘‡βŸΆ(𝒫 𝑆 ∩ Fin) ∧ 𝑇 βŠ† (π‘β€˜βˆͺ ran 𝑓))) β†’ 𝑇 βŠ† (π‘β€˜βˆͺ ran 𝑓))
1913, 2mrcssvd 17571 . . . . . . . . . 10 ((πœ‘ ∧ (𝑓:π‘‡βŸΆ(𝒫 𝑆 ∩ Fin) ∧ 𝑇 βŠ† (π‘β€˜βˆͺ ran 𝑓))) β†’ (π‘β€˜βˆͺ ran 𝑓) βŠ† 𝑋)
2013, 2, 18, 19mrcssd 17572 . . . . . . . . 9 ((πœ‘ ∧ (𝑓:π‘‡βŸΆ(𝒫 𝑆 ∩ Fin) ∧ 𝑇 βŠ† (π‘β€˜βˆͺ ran 𝑓))) β†’ (π‘β€˜π‘‡) βŠ† (π‘β€˜(π‘β€˜βˆͺ ran 𝑓)))
21 frn 6724 . . . . . . . . . . . . . 14 (𝑓:π‘‡βŸΆ(𝒫 𝑆 ∩ Fin) β†’ ran 𝑓 βŠ† (𝒫 𝑆 ∩ Fin))
2221unissd 4918 . . . . . . . . . . . . 13 (𝑓:π‘‡βŸΆ(𝒫 𝑆 ∩ Fin) β†’ βˆͺ ran 𝑓 βŠ† βˆͺ (𝒫 𝑆 ∩ Fin))
23 unifpw 9357 . . . . . . . . . . . . 13 βˆͺ (𝒫 𝑆 ∩ Fin) = 𝑆
2422, 23sseqtrdi 4032 . . . . . . . . . . . 12 (𝑓:π‘‡βŸΆ(𝒫 𝑆 ∩ Fin) β†’ βˆͺ ran 𝑓 βŠ† 𝑆)
2524ad2antrl 726 . . . . . . . . . . 11 ((πœ‘ ∧ (𝑓:π‘‡βŸΆ(𝒫 𝑆 ∩ Fin) ∧ 𝑇 βŠ† (π‘β€˜βˆͺ ran 𝑓))) β†’ βˆͺ ran 𝑓 βŠ† 𝑆)
2625, 15sstrd 3992 . . . . . . . . . 10 ((πœ‘ ∧ (𝑓:π‘‡βŸΆ(𝒫 𝑆 ∩ Fin) ∧ 𝑇 βŠ† (π‘β€˜βˆͺ ran 𝑓))) β†’ βˆͺ ran 𝑓 βŠ† 𝑋)
2713, 2, 26mrcidmd 17574 . . . . . . . . 9 ((πœ‘ ∧ (𝑓:π‘‡βŸΆ(𝒫 𝑆 ∩ Fin) ∧ 𝑇 βŠ† (π‘β€˜βˆͺ ran 𝑓))) β†’ (π‘β€˜(π‘β€˜βˆͺ ran 𝑓)) = (π‘β€˜βˆͺ ran 𝑓))
2820, 27sseqtrd 4022 . . . . . . . 8 ((πœ‘ ∧ (𝑓:π‘‡βŸΆ(𝒫 𝑆 ∩ Fin) ∧ 𝑇 βŠ† (π‘β€˜βˆͺ ran 𝑓))) β†’ (π‘β€˜π‘‡) βŠ† (π‘β€˜βˆͺ ran 𝑓))
2917, 28eqsstrd 4020 . . . . . . 7 ((πœ‘ ∧ (𝑓:π‘‡βŸΆ(𝒫 𝑆 ∩ Fin) ∧ 𝑇 βŠ† (π‘β€˜βˆͺ ran 𝑓))) β†’ (π‘β€˜π‘†) βŠ† (π‘β€˜βˆͺ ran 𝑓))
3016, 29sstrd 3992 . . . . . 6 ((πœ‘ ∧ (𝑓:π‘‡βŸΆ(𝒫 𝑆 ∩ Fin) ∧ 𝑇 βŠ† (π‘β€˜βˆͺ ran 𝑓))) β†’ 𝑆 βŠ† (π‘β€˜βˆͺ ran 𝑓))
3113, 2, 3, 30, 25, 14mrissmrcd 17588 . . . . 5 ((πœ‘ ∧ (𝑓:π‘‡βŸΆ(𝒫 𝑆 ∩ Fin) ∧ 𝑇 βŠ† (π‘β€˜βˆͺ ran 𝑓))) β†’ 𝑆 = βˆͺ ran 𝑓)
3212, 31jca 512 . . . 4 ((πœ‘ ∧ (𝑓:π‘‡βŸΆ(𝒫 𝑆 ∩ Fin) ∧ 𝑇 βŠ† (π‘β€˜βˆͺ ran 𝑓))) β†’ (𝑓:π‘‡βŸΆ(𝒫 𝑆 ∩ Fin) ∧ 𝑆 = βˆͺ ran 𝑓))
3332ex 413 . . 3 (πœ‘ β†’ ((𝑓:π‘‡βŸΆ(𝒫 𝑆 ∩ Fin) ∧ 𝑇 βŠ† (π‘β€˜βˆͺ ran 𝑓)) β†’ (𝑓:π‘‡βŸΆ(𝒫 𝑆 ∩ Fin) ∧ 𝑆 = βˆͺ ran 𝑓)))
3433eximdv 1920 . 2 (πœ‘ β†’ (βˆƒπ‘“(𝑓:π‘‡βŸΆ(𝒫 𝑆 ∩ Fin) ∧ 𝑇 βŠ† (π‘β€˜βˆͺ ran 𝑓)) β†’ βˆƒπ‘“(𝑓:π‘‡βŸΆ(𝒫 𝑆 ∩ Fin) ∧ 𝑆 = βˆͺ ran 𝑓)))
3511, 34mpd 15 1 (πœ‘ β†’ βˆƒπ‘“(𝑓:π‘‡βŸΆ(𝒫 𝑆 ∩ Fin) ∧ 𝑆 = βˆͺ ran 𝑓))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   = wceq 1541  βˆƒwex 1781   ∈ wcel 2106   ∩ cin 3947   βŠ† wss 3948  π’« cpw 4602  βˆͺ cuni 4908  ran crn 5677  βŸΆwf 6539  β€˜cfv 6543  Fincfn 8941  Moorecmre 17530  mrClscmrc 17531  mrIndcmri 17532  ACScacs 17533
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 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727  ax-reg 9589  ax-inf2 9638  ax-ac2 10460  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189
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 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-iin 5000  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-se 5632  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-r1 9761  df-rank 9762  df-card 9936  df-ac 10113  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-nn 12217  df-2 12279  df-3 12280  df-4 12281  df-5 12282  df-6 12283  df-7 12284  df-8 12285  df-9 12286  df-n0 12477  df-z 12563  df-dec 12682  df-uz 12827  df-fz 13489  df-struct 17084  df-slot 17119  df-ndx 17131  df-base 17149  df-tset 17220  df-ple 17221  df-ocomp 17222  df-mre 17534  df-mrc 17535  df-mri 17536  df-acs 17537  df-proset 18252  df-drs 18253  df-poset 18270  df-ipo 18485
This theorem is referenced by:  acsinfd  18513  acsdomd  18514
  Copyright terms: Public domain W3C validator