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Theorem acsmapd 18451
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 18449 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 6859 . . . . 5 (π‘β€˜π‘†) ∈ V
32ssex 5282 . . . 4 (𝑇 βŠ† (π‘β€˜π‘†) β†’ 𝑇 ∈ V)
41, 3syl 17 . . 3 (πœ‘ β†’ 𝑇 ∈ V)
51sseld 3947 . . . . 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 18449 . . . . 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 3139 . . 3 (πœ‘ β†’ βˆ€π‘₯ ∈ 𝑇 βˆƒπ‘¦ ∈ (𝒫 𝑆 ∩ Fin)π‘₯ ∈ (π‘β€˜π‘¦))
12 fveq2 6846 . . . . 5 (𝑦 = (π‘“β€˜π‘₯) β†’ (π‘β€˜π‘¦) = (π‘β€˜(π‘“β€˜π‘₯)))
1312eleq2d 2820 . . . 4 (𝑦 = (π‘“β€˜π‘₯) β†’ (π‘₯ ∈ (π‘β€˜π‘¦) ↔ π‘₯ ∈ (π‘β€˜(π‘“β€˜π‘₯))))
1413ac6sg 10432 . . 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 770 . . . . 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 1918 . . . . . . . 8 β„²π‘₯πœ‘
18 nfv 1918 . . . . . . . . 9 β„²π‘₯ 𝑓:π‘‡βŸΆ(𝒫 𝑆 ∩ Fin)
19 nfra1 3266 . . . . . . . . 9 β„²π‘₯βˆ€π‘₯ ∈ 𝑇 π‘₯ ∈ (π‘β€˜(π‘“β€˜π‘₯))
2018, 19nfan 1903 . . . . . . . 8 β„²π‘₯(𝑓:π‘‡βŸΆ(𝒫 𝑆 ∩ Fin) ∧ βˆ€π‘₯ ∈ 𝑇 π‘₯ ∈ (π‘β€˜(π‘“β€˜π‘₯)))
2117, 20nfan 1903 . . . . . . 7 β„²π‘₯(πœ‘ ∧ (𝑓:π‘‡βŸΆ(𝒫 𝑆 ∩ Fin) ∧ βˆ€π‘₯ ∈ 𝑇 π‘₯ ∈ (π‘β€˜(π‘“β€˜π‘₯))))
226ad2antrr 725 . . . . . . . . . . 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 17544 . . . . . . . . . 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 776 . . . . . . . . . . . . . 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 6673 . . . . . . . . . . . . 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 7035 . . . . . . . . . . . . 13 ((𝑓 Fn 𝑇 ∧ π‘₯ ∈ 𝑇) β†’ (π‘“β€˜π‘₯) ∈ ran 𝑓)
2725, 26sylancom 589 . . . . . . . . . . . 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 4773 . . . . . . . . . . 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 4879 . . . . . . . . . 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 6679 . . . . . . . . . . . . . 14 (𝑓:π‘‡βŸΆ(𝒫 𝑆 ∩ Fin) β†’ ran 𝑓 βŠ† (𝒫 𝑆 ∩ Fin))
3130unissd 4879 . . . . . . . . . . . . 13 (𝑓:π‘‡βŸΆ(𝒫 𝑆 ∩ Fin) β†’ βˆͺ ran 𝑓 βŠ† βˆͺ (𝒫 𝑆 ∩ Fin))
32 unifpw 9305 . . . . . . . . . . . . 13 βˆͺ (𝒫 𝑆 ∩ Fin) = 𝑆
3331, 32sseqtrdi 3998 . . . . . . . . . . . 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 725 . . . . . . . . . . 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 3958 . . . . . . . . . 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 17512 . . . . . . . . 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 772 . . . . . . . . . . 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 3233 . . . . . . . . . 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 6859 . . . . . . . . . . . 12 (π‘“β€˜π‘₯) ∈ V
4140unisn 4891 . . . . . . . . . . 11 βˆͺ {(π‘“β€˜π‘₯)} = (π‘“β€˜π‘₯)
4241fveq2i 6849 . . . . . . . . . 10 (π‘β€˜βˆͺ {(π‘“β€˜π‘₯)}) = (π‘β€˜(π‘“β€˜π‘₯))
4339, 42eleqtrrdi 2845 . . . . . . . . 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 3949 . . . . . . . 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 414 . . . . . . 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 2207 . . . . . 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 3934 . . . . . 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 513 . . . 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 414 . . 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 1921 . 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 397  βˆ€wal 1540   = wceq 1542  βˆƒwex 1782   ∈ wcel 2107  βˆ€wral 3061  βˆƒwrex 3070  Vcvv 3447   ∩ cin 3913   βŠ† wss 3914  π’« cpw 4564  {csn 4590  βˆͺ cuni 4869  ran crn 5638   Fn wfn 6495  βŸΆwf 6496  β€˜cfv 6500  Fincfn 8889  mrClscmrc 17471  ACScacs 17473
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676  ax-reg 9536  ax-inf2 9585  ax-ac2 10407  ax-cnex 11115  ax-resscn 11116  ax-1cn 11117  ax-icn 11118  ax-addcl 11119  ax-addrcl 11120  ax-mulcl 11121  ax-mulrcl 11122  ax-mulcom 11123  ax-addass 11124  ax-mulass 11125  ax-distr 11126  ax-i2m1 11127  ax-1ne0 11128  ax-1rid 11129  ax-rnegex 11130  ax-rrecex 11131  ax-cnre 11132  ax-pre-lttri 11133  ax-pre-lttrn 11134  ax-pre-ltadd 11135  ax-pre-mulgt0 11136
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-int 4912  df-iun 4960  df-iin 4961  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-se 5593  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-pred 6257  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-isom 6509  df-riota 7317  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7807  df-1st 7925  df-2nd 7926  df-frecs 8216  df-wrecs 8247  df-recs 8321  df-rdg 8360  df-1o 8416  df-er 8654  df-en 8890  df-dom 8891  df-sdom 8892  df-fin 8893  df-r1 9708  df-rank 9709  df-card 9883  df-ac 10060  df-pnf 11199  df-mnf 11200  df-xr 11201  df-ltxr 11202  df-le 11203  df-sub 11395  df-neg 11396  df-nn 12162  df-2 12224  df-3 12225  df-4 12226  df-5 12227  df-6 12228  df-7 12229  df-8 12230  df-9 12231  df-n0 12422  df-z 12508  df-dec 12627  df-uz 12772  df-fz 13434  df-struct 17027  df-slot 17062  df-ndx 17074  df-base 17092  df-tset 17160  df-ple 17161  df-ocomp 17162  df-mre 17474  df-mrc 17475  df-acs 17477  df-proset 18192  df-drs 18193  df-poset 18210  df-ipo 18425
This theorem is referenced by:  acsmap2d  18452
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