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Theorem acsmapd 18553
Description: In an algebraic closure system, if 𝑇 is contained in the closure of 𝑆, there is a map 𝑓 from 𝑇 into the set of finite subsets of 𝑆 such that the closure of βˆͺ ran 𝑓 contains 𝑇. This is proven by applying acsficl2d 18551 to each element of 𝑇. See Section II.5 in [Cohn] p. 81 to 82. (Contributed by David Moews, 1-May-2017.)
Hypotheses
Ref Expression
acsmapd.1 (πœ‘ β†’ 𝐴 ∈ (ACSβ€˜π‘‹))
acsmapd.2 𝑁 = (mrClsβ€˜π΄)
acsmapd.3 (πœ‘ β†’ 𝑆 βŠ† 𝑋)
acsmapd.4 (πœ‘ β†’ 𝑇 βŠ† (π‘β€˜π‘†))
Assertion
Ref Expression
acsmapd (πœ‘ β†’ βˆƒπ‘“(𝑓:π‘‡βŸΆ(𝒫 𝑆 ∩ Fin) ∧ 𝑇 βŠ† (π‘β€˜βˆͺ ran 𝑓)))
Distinct variable groups:   𝑇,𝑓   πœ‘,𝑓   𝑆,𝑓   𝑓,𝑁
Allowed substitution hints:   𝐴(𝑓)   𝑋(𝑓)
Proof of Theorem acsmapd
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 acsmapd.4 . . . 4 (πœ‘ β†’ 𝑇 βŠ† (π‘β€˜π‘†))
2 fvex 6915 . . . . 5 (π‘β€˜π‘†) ∈ V
32ssex 5325 . . . 4 (𝑇 βŠ† (π‘β€˜π‘†) β†’ 𝑇 ∈ V)
41, 3syl 17 . . 3 (πœ‘ β†’ 𝑇 ∈ V)
51sseld 3981 . . . . 5 (πœ‘ β†’ (π‘₯ ∈ 𝑇 β†’ π‘₯ ∈ (π‘β€˜π‘†)))
6 acsmapd.1 . . . . . 6 (πœ‘ β†’ 𝐴 ∈ (ACSβ€˜π‘‹))
7 acsmapd.2 . . . . . 6 𝑁 = (mrClsβ€˜π΄)
8 acsmapd.3 . . . . . 6 (πœ‘ β†’ 𝑆 βŠ† 𝑋)
96, 7, 8acsficl2d 18551 . . . . 5 (πœ‘ β†’ (π‘₯ ∈ (π‘β€˜π‘†) ↔ βˆƒπ‘¦ ∈ (𝒫 𝑆 ∩ Fin)π‘₯ ∈ (π‘β€˜π‘¦)))
105, 9sylibd 238 . . . 4 (πœ‘ β†’ (π‘₯ ∈ 𝑇 β†’ βˆƒπ‘¦ ∈ (𝒫 𝑆 ∩ Fin)π‘₯ ∈ (π‘β€˜π‘¦)))
1110ralrimiv 3142 . . 3 (πœ‘ β†’ βˆ€π‘₯ ∈ 𝑇 βˆƒπ‘¦ ∈ (𝒫 𝑆 ∩ Fin)π‘₯ ∈ (π‘β€˜π‘¦))
12 fveq2 6902 . . . . 5 (𝑦 = (π‘“β€˜π‘₯) β†’ (π‘β€˜π‘¦) = (π‘β€˜(π‘“β€˜π‘₯)))
1312eleq2d 2815 . . . 4 (𝑦 = (π‘“β€˜π‘₯) β†’ (π‘₯ ∈ (π‘β€˜π‘¦) ↔ π‘₯ ∈ (π‘β€˜(π‘“β€˜π‘₯))))
1413ac6sg 10519 . . 3 (𝑇 ∈ V β†’ (βˆ€π‘₯ ∈ 𝑇 βˆƒπ‘¦ ∈ (𝒫 𝑆 ∩ Fin)π‘₯ ∈ (π‘β€˜π‘¦) β†’ βˆƒπ‘“(𝑓:π‘‡βŸΆ(𝒫 𝑆 ∩ Fin) ∧ βˆ€π‘₯ ∈ 𝑇 π‘₯ ∈ (π‘β€˜(π‘“β€˜π‘₯)))))
154, 11, 14sylc 65 . 2 (πœ‘ β†’ βˆƒπ‘“(𝑓:π‘‡βŸΆ(𝒫 𝑆 ∩ Fin) ∧ βˆ€π‘₯ ∈ 𝑇 π‘₯ ∈ (π‘β€˜(π‘“β€˜π‘₯))))
16 simprl 769 . . . . 5 ((πœ‘ ∧ (𝑓:π‘‡βŸΆ(𝒫 𝑆 ∩ Fin) ∧ βˆ€π‘₯ ∈ 𝑇 π‘₯ ∈ (π‘β€˜(π‘“β€˜π‘₯)))) β†’ 𝑓:π‘‡βŸΆ(𝒫 𝑆 ∩ Fin))
17 nfv 1909 . . . . . . . 8 β„²π‘₯πœ‘
18 nfv 1909 . . . . . . . . 9 β„²π‘₯ 𝑓:π‘‡βŸΆ(𝒫 𝑆 ∩ Fin)
19 nfra1 3279 . . . . . . . . 9 β„²π‘₯βˆ€π‘₯ ∈ 𝑇 π‘₯ ∈ (π‘β€˜(π‘“β€˜π‘₯))
2018, 19nfan 1894 . . . . . . . 8 β„²π‘₯(𝑓:π‘‡βŸΆ(𝒫 𝑆 ∩ Fin) ∧ βˆ€π‘₯ ∈ 𝑇 π‘₯ ∈ (π‘β€˜(π‘“β€˜π‘₯)))
2117, 20nfan 1894 . . . . . . 7 β„²π‘₯(πœ‘ ∧ (𝑓:π‘‡βŸΆ(𝒫 𝑆 ∩ Fin) ∧ βˆ€π‘₯ ∈ 𝑇 π‘₯ ∈ (π‘β€˜(π‘“β€˜π‘₯))))
226ad2antrr 724 . . . . . . . . . . 11 (((πœ‘ ∧ (𝑓:π‘‡βŸΆ(𝒫 𝑆 ∩ Fin) ∧ βˆ€π‘₯ ∈ 𝑇 π‘₯ ∈ (π‘β€˜(π‘“β€˜π‘₯)))) ∧ π‘₯ ∈ 𝑇) β†’ 𝐴 ∈ (ACSβ€˜π‘‹))
2322acsmred 17643 . . . . . . . . . 10 (((πœ‘ ∧ (𝑓:π‘‡βŸΆ(𝒫 𝑆 ∩ Fin) ∧ βˆ€π‘₯ ∈ 𝑇 π‘₯ ∈ (π‘β€˜(π‘“β€˜π‘₯)))) ∧ π‘₯ ∈ 𝑇) β†’ 𝐴 ∈ (Mooreβ€˜π‘‹))
24 simplrl 775 . . . . . . . . . . . . . 14 (((πœ‘ ∧ (𝑓:π‘‡βŸΆ(𝒫 𝑆 ∩ Fin) ∧ βˆ€π‘₯ ∈ 𝑇 π‘₯ ∈ (π‘β€˜(π‘“β€˜π‘₯)))) ∧ π‘₯ ∈ 𝑇) β†’ 𝑓:π‘‡βŸΆ(𝒫 𝑆 ∩ Fin))
2524ffnd 6728 . . . . . . . . . . . . 13 (((πœ‘ ∧ (𝑓:π‘‡βŸΆ(𝒫 𝑆 ∩ Fin) ∧ βˆ€π‘₯ ∈ 𝑇 π‘₯ ∈ (π‘β€˜(π‘“β€˜π‘₯)))) ∧ π‘₯ ∈ 𝑇) β†’ 𝑓 Fn 𝑇)
26 fnfvelrn 7095 . . . . . . . . . . . . 13 ((𝑓 Fn 𝑇 ∧ π‘₯ ∈ 𝑇) β†’ (π‘“β€˜π‘₯) ∈ ran 𝑓)
2725, 26sylancom 586 . . . . . . . . . . . 12 (((πœ‘ ∧ (𝑓:π‘‡βŸΆ(𝒫 𝑆 ∩ Fin) ∧ βˆ€π‘₯ ∈ 𝑇 π‘₯ ∈ (π‘β€˜(π‘“β€˜π‘₯)))) ∧ π‘₯ ∈ 𝑇) β†’ (π‘“β€˜π‘₯) ∈ ran 𝑓)
2827snssd 4817 . . . . . . . . . . 11 (((πœ‘ ∧ (𝑓:π‘‡βŸΆ(𝒫 𝑆 ∩ Fin) ∧ βˆ€π‘₯ ∈ 𝑇 π‘₯ ∈ (π‘β€˜(π‘“β€˜π‘₯)))) ∧ π‘₯ ∈ 𝑇) β†’ {(π‘“β€˜π‘₯)} βŠ† ran 𝑓)
2928unissd 4922 . . . . . . . . . 10 (((πœ‘ ∧ (𝑓:π‘‡βŸΆ(𝒫 𝑆 ∩ Fin) ∧ βˆ€π‘₯ ∈ 𝑇 π‘₯ ∈ (π‘β€˜(π‘“β€˜π‘₯)))) ∧ π‘₯ ∈ 𝑇) β†’ βˆͺ {(π‘“β€˜π‘₯)} βŠ† βˆͺ ran 𝑓)
30 frn 6734 . . . . . . . . . . . . . 14 (𝑓:π‘‡βŸΆ(𝒫 𝑆 ∩ Fin) β†’ ran 𝑓 βŠ† (𝒫 𝑆 ∩ Fin))
3130unissd 4922 . . . . . . . . . . . . 13 (𝑓:π‘‡βŸΆ(𝒫 𝑆 ∩ Fin) β†’ βˆͺ ran 𝑓 βŠ† βˆͺ (𝒫 𝑆 ∩ Fin))
32 unifpw 9387 . . . . . . . . . . . . 13 βˆͺ (𝒫 𝑆 ∩ Fin) = 𝑆
3331, 32sseqtrdi 4032 . . . . . . . . . . . 12 (𝑓:π‘‡βŸΆ(𝒫 𝑆 ∩ Fin) β†’ βˆͺ ran 𝑓 βŠ† 𝑆)
3424, 33syl 17 . . . . . . . . . . 11 (((πœ‘ ∧ (𝑓:π‘‡βŸΆ(𝒫 𝑆 ∩ Fin) ∧ βˆ€π‘₯ ∈ 𝑇 π‘₯ ∈ (π‘β€˜(π‘“β€˜π‘₯)))) ∧ π‘₯ ∈ 𝑇) β†’ βˆͺ ran 𝑓 βŠ† 𝑆)
358ad2antrr 724 . . . . . . . . . . 11 (((πœ‘ ∧ (𝑓:π‘‡βŸΆ(𝒫 𝑆 ∩ Fin) ∧ βˆ€π‘₯ ∈ 𝑇 π‘₯ ∈ (π‘β€˜(π‘“β€˜π‘₯)))) ∧ π‘₯ ∈ 𝑇) β†’ 𝑆 βŠ† 𝑋)
3634, 35sstrd 3992 . . . . . . . . . 10 (((πœ‘ ∧ (𝑓:π‘‡βŸΆ(𝒫 𝑆 ∩ Fin) ∧ βˆ€π‘₯ ∈ 𝑇 π‘₯ ∈ (π‘β€˜(π‘“β€˜π‘₯)))) ∧ π‘₯ ∈ 𝑇) β†’ βˆͺ ran 𝑓 βŠ† 𝑋)
3723, 7, 29, 36mrcssd 17611 . . . . . . . . 9 (((πœ‘ ∧ (𝑓:π‘‡βŸΆ(𝒫 𝑆 ∩ Fin) ∧ βˆ€π‘₯ ∈ 𝑇 π‘₯ ∈ (π‘β€˜(π‘“β€˜π‘₯)))) ∧ π‘₯ ∈ 𝑇) β†’ (π‘β€˜βˆͺ {(π‘“β€˜π‘₯)}) βŠ† (π‘β€˜βˆͺ ran 𝑓))
38 simprr 771 . . . . . . . . . . 11 ((πœ‘ ∧ (𝑓:π‘‡βŸΆ(𝒫 𝑆 ∩ Fin) ∧ βˆ€π‘₯ ∈ 𝑇 π‘₯ ∈ (π‘β€˜(π‘“β€˜π‘₯)))) β†’ βˆ€π‘₯ ∈ 𝑇 π‘₯ ∈ (π‘β€˜(π‘“β€˜π‘₯)))
3938r19.21bi 3246 . . . . . . . . . 10 (((πœ‘ ∧ (𝑓:π‘‡βŸΆ(𝒫 𝑆 ∩ Fin) ∧ βˆ€π‘₯ ∈ 𝑇 π‘₯ ∈ (π‘β€˜(π‘“β€˜π‘₯)))) ∧ π‘₯ ∈ 𝑇) β†’ π‘₯ ∈ (π‘β€˜(π‘“β€˜π‘₯)))
40 fvex 6915 . . . . . . . . . . . 12 (π‘“β€˜π‘₯) ∈ V
4140unisn 4933 . . . . . . . . . . 11 βˆͺ {(π‘“β€˜π‘₯)} = (π‘“β€˜π‘₯)
4241fveq2i 6905 . . . . . . . . . 10 (π‘β€˜βˆͺ {(π‘“β€˜π‘₯)}) = (π‘β€˜(π‘“β€˜π‘₯))
4339, 42eleqtrrdi 2840 . . . . . . . . 9 (((πœ‘ ∧ (𝑓:π‘‡βŸΆ(𝒫 𝑆 ∩ Fin) ∧ βˆ€π‘₯ ∈ 𝑇 π‘₯ ∈ (π‘β€˜(π‘“β€˜π‘₯)))) ∧ π‘₯ ∈ 𝑇) β†’ π‘₯ ∈ (π‘β€˜βˆͺ {(π‘“β€˜π‘₯)}))
4437, 43sseldd 3983 . . . . . . . 8 (((πœ‘ ∧ (𝑓:π‘‡βŸΆ(𝒫 𝑆 ∩ Fin) ∧ βˆ€π‘₯ ∈ 𝑇 π‘₯ ∈ (π‘β€˜(π‘“β€˜π‘₯)))) ∧ π‘₯ ∈ 𝑇) β†’ π‘₯ ∈ (π‘β€˜βˆͺ ran 𝑓))
4544ex 411 . . . . . . 7 ((πœ‘ ∧ (𝑓:π‘‡βŸΆ(𝒫 𝑆 ∩ Fin) ∧ βˆ€π‘₯ ∈ 𝑇 π‘₯ ∈ (π‘β€˜(π‘“β€˜π‘₯)))) β†’ (π‘₯ ∈ 𝑇 β†’ π‘₯ ∈ (π‘β€˜βˆͺ ran 𝑓)))
4621, 45alrimi 2201 . . . . . 6 ((πœ‘ ∧ (𝑓:π‘‡βŸΆ(𝒫 𝑆 ∩ Fin) ∧ βˆ€π‘₯ ∈ 𝑇 π‘₯ ∈ (π‘β€˜(π‘“β€˜π‘₯)))) β†’ βˆ€π‘₯(π‘₯ ∈ 𝑇 β†’ π‘₯ ∈ (π‘β€˜βˆͺ ran 𝑓)))
47 dfss2 3969 . . . . . 6 (𝑇 βŠ† (π‘β€˜βˆͺ ran 𝑓) ↔ βˆ€π‘₯(π‘₯ ∈ 𝑇 β†’ π‘₯ ∈ (π‘β€˜βˆͺ ran 𝑓)))
4846, 47sylibr 233 . . . . 5 ((πœ‘ ∧ (𝑓:π‘‡βŸΆ(𝒫 𝑆 ∩ Fin) ∧ βˆ€π‘₯ ∈ 𝑇 π‘₯ ∈ (π‘β€˜(π‘“β€˜π‘₯)))) β†’ 𝑇 βŠ† (π‘β€˜βˆͺ ran 𝑓))
4916, 48jca 510 . . . 4 ((πœ‘ ∧ (𝑓:π‘‡βŸΆ(𝒫 𝑆 ∩ Fin) ∧ βˆ€π‘₯ ∈ 𝑇 π‘₯ ∈ (π‘β€˜(π‘“β€˜π‘₯)))) β†’ (𝑓:π‘‡βŸΆ(𝒫 𝑆 ∩ Fin) ∧ 𝑇 βŠ† (π‘β€˜βˆͺ ran 𝑓)))
5049ex 411 . . 3 (πœ‘ β†’ ((𝑓:π‘‡βŸΆ(𝒫 𝑆 ∩ Fin) ∧ βˆ€π‘₯ ∈ 𝑇 π‘₯ ∈ (π‘β€˜(π‘“β€˜π‘₯))) β†’ (𝑓:π‘‡βŸΆ(𝒫 𝑆 ∩ Fin) ∧ 𝑇 βŠ† (π‘β€˜βˆͺ ran 𝑓))))
5150eximdv 1912 . 2 (πœ‘ β†’ (βˆƒπ‘“(𝑓:π‘‡βŸΆ(𝒫 𝑆 ∩ Fin) ∧ βˆ€π‘₯ ∈ 𝑇 π‘₯ ∈ (π‘β€˜(π‘“β€˜π‘₯))) β†’ βˆƒπ‘“(𝑓:π‘‡βŸΆ(𝒫 𝑆 ∩ Fin) ∧ 𝑇 βŠ† (π‘β€˜βˆͺ ran 𝑓))))
5215, 51mpd 15 1 (πœ‘ β†’ βˆƒπ‘“(𝑓:π‘‡βŸΆ(𝒫 𝑆 ∩ Fin) ∧ 𝑇 βŠ† (π‘β€˜βˆͺ ran 𝑓)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394  βˆ€wal 1531   = wceq 1533  βˆƒwex 1773   ∈ wcel 2098  βˆ€wral 3058  βˆƒwrex 3067  Vcvv 3473   ∩ cin 3948   βŠ† wss 3949  π’« cpw 4606  {csn 4632  βˆͺ cuni 4912  ran crn 5683   Fn wfn 6548  βŸΆwf 6549  β€˜cfv 6553  Fincfn 8970  mrClscmrc 17570  ACScacs 17572
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-reg 9623  ax-inf2 9672  ax-ac2 10494  ax-cnex 11202  ax-resscn 11203  ax-1cn 11204  ax-icn 11205  ax-addcl 11206  ax-addrcl 11207  ax-mulcl 11208  ax-mulrcl 11209  ax-mulcom 11210  ax-addass 11211  ax-mulass 11212  ax-distr 11213  ax-i2m1 11214  ax-1ne0 11215  ax-1rid 11216  ax-rnegex 11217  ax-rrecex 11218  ax-cnre 11219  ax-pre-lttri 11220  ax-pre-lttrn 11221  ax-pre-ltadd 11222  ax-pre-mulgt0 11223
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-int 4954  df-iun 5002  df-iin 5003  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-se 5638  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6310  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-isom 6562  df-riota 7382  df-ov 7429  df-oprab 7430  df-mpo 7431  df-om 7877  df-1st 7999  df-2nd 8000  df-frecs 8293  df-wrecs 8324  df-recs 8398  df-rdg 8437  df-1o 8493  df-er 8731  df-en 8971  df-dom 8972  df-sdom 8973  df-fin 8974  df-r1 9795  df-rank 9796  df-card 9970  df-ac 10147  df-pnf 11288  df-mnf 11289  df-xr 11290  df-ltxr 11291  df-le 11292  df-sub 11484  df-neg 11485  df-nn 12251  df-2 12313  df-3 12314  df-4 12315  df-5 12316  df-6 12317  df-7 12318  df-8 12319  df-9 12320  df-n0 12511  df-z 12597  df-dec 12716  df-uz 12861  df-fz 13525  df-struct 17123  df-slot 17158  df-ndx 17170  df-base 17188  df-tset 17259  df-ple 17260  df-ocomp 17261  df-mre 17573  df-mrc 17574  df-acs 17576  df-proset 18294  df-drs 18295  df-poset 18312  df-ipo 18527
This theorem is referenced by:  acsmap2d  18554
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