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Theorem acsmap2d 14607
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 14606 and observing that, since  S and  T have the same closure, the closure of  U. ran  f must contain  S. Since  S is independent, by mrissmrcd 13867,  U. 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  |-  ( ph  ->  A  e.  (ACS `  X ) )
acsmap2d.2  |-  N  =  (mrCls `  A )
acsmap2d.3  |-  I  =  (mrInd `  A )
acsmap2d.4  |-  ( ph  ->  S  e.  I )
acsmap2d.5  |-  ( ph  ->  T  C_  X )
acsmap2d.6  |-  ( ph  ->  ( N `  S
)  =  ( N `
 T ) )
Assertion
Ref Expression
acsmap2d  |-  ( ph  ->  E. f ( f : T --> ( ~P S  i^i  Fin )  /\  S  =  U. ran  f ) )
Distinct variable groups:    S, f    T, f    ph, f    f, N
Allowed substitution hints:    A( f)    I(
f)    X( f)

Proof of Theorem acsmap2d
StepHypRef Expression
1 acsmap2d.1 . . 3  |-  ( ph  ->  A  e.  (ACS `  X ) )
2 acsmap2d.2 . . 3  |-  N  =  (mrCls `  A )
3 acsmap2d.3 . . . 4  |-  I  =  (mrInd `  A )
41acsmred 13883 . . . 4  |-  ( ph  ->  A  e.  (Moore `  X ) )
5 acsmap2d.4 . . . 4  |-  ( ph  ->  S  e.  I )
63, 4, 5mrissd 13863 . . 3  |-  ( ph  ->  S  C_  X )
7 acsmap2d.5 . . . . 5  |-  ( ph  ->  T  C_  X )
84, 2, 7mrcssidd 13852 . . . 4  |-  ( ph  ->  T  C_  ( N `  T ) )
9 acsmap2d.6 . . . 4  |-  ( ph  ->  ( N `  S
)  =  ( N `
 T ) )
108, 9sseqtr4d 3387 . . 3  |-  ( ph  ->  T  C_  ( N `  S ) )
111, 2, 6, 10acsmapd 14606 . 2  |-  ( ph  ->  E. f ( f : T --> ( ~P S  i^i  Fin )  /\  T  C_  ( N `
 U. ran  f
) ) )
12 simprl 734 . . . . 5  |-  ( (
ph  /\  ( f : T --> ( ~P S  i^i  Fin )  /\  T  C_  ( N `  U. ran  f ) ) )  ->  f : T --> ( ~P S  i^i  Fin ) )
134adantr 453 . . . . . 6  |-  ( (
ph  /\  ( f : T --> ( ~P S  i^i  Fin )  /\  T  C_  ( N `  U. ran  f ) ) )  ->  A  e.  (Moore `  X ) )
145adantr 453 . . . . . . . . 9  |-  ( (
ph  /\  ( f : T --> ( ~P S  i^i  Fin )  /\  T  C_  ( N `  U. ran  f ) ) )  ->  S  e.  I
)
153, 13, 14mrissd 13863 . . . . . . . 8  |-  ( (
ph  /\  ( f : T --> ( ~P S  i^i  Fin )  /\  T  C_  ( N `  U. ran  f ) ) )  ->  S  C_  X
)
1613, 2, 15mrcssidd 13852 . . . . . . 7  |-  ( (
ph  /\  ( f : T --> ( ~P S  i^i  Fin )  /\  T  C_  ( N `  U. ran  f ) ) )  ->  S  C_  ( N `  S )
)
179adantr 453 . . . . . . . 8  |-  ( (
ph  /\  ( f : T --> ( ~P S  i^i  Fin )  /\  T  C_  ( N `  U. ran  f ) ) )  ->  ( N `  S )  =  ( N `  T ) )
18 simprr 735 . . . . . . . . . 10  |-  ( (
ph  /\  ( f : T --> ( ~P S  i^i  Fin )  /\  T  C_  ( N `  U. ran  f ) ) )  ->  T  C_  ( N `  U. ran  f
) )
1913, 2mrcssvd 13850 . . . . . . . . . 10  |-  ( (
ph  /\  ( f : T --> ( ~P S  i^i  Fin )  /\  T  C_  ( N `  U. ran  f ) ) )  ->  ( N `  U. ran  f )  C_  X )
2013, 2, 18, 19mrcssd 13851 . . . . . . . . 9  |-  ( (
ph  /\  ( f : T --> ( ~P S  i^i  Fin )  /\  T  C_  ( N `  U. ran  f ) ) )  ->  ( N `  T )  C_  ( N `  ( N `  U. ran  f ) ) )
21 frn 5599 . . . . . . . . . . . . . 14  |-  ( f : T --> ( ~P S  i^i  Fin )  ->  ran  f  C_  ( ~P S  i^i  Fin )
)
2221unissd 4041 . . . . . . . . . . . . 13  |-  ( f : T --> ( ~P S  i^i  Fin )  ->  U. ran  f  C_  U. ( ~P S  i^i  Fin ) )
23 unifpw 7411 . . . . . . . . . . . . 13  |-  U. ( ~P S  i^i  Fin )  =  S
2422, 23syl6sseq 3396 . . . . . . . . . . . 12  |-  ( f : T --> ( ~P S  i^i  Fin )  ->  U. ran  f  C_  S )
2524ad2antrl 710 . . . . . . . . . . 11  |-  ( (
ph  /\  ( f : T --> ( ~P S  i^i  Fin )  /\  T  C_  ( N `  U. ran  f ) ) )  ->  U. ran  f  C_  S )
2625, 15sstrd 3360 . . . . . . . . . 10  |-  ( (
ph  /\  ( f : T --> ( ~P S  i^i  Fin )  /\  T  C_  ( N `  U. ran  f ) ) )  ->  U. ran  f  C_  X )
2713, 2, 26mrcidmd 13853 . . . . . . . . 9  |-  ( (
ph  /\  ( f : T --> ( ~P S  i^i  Fin )  /\  T  C_  ( N `  U. ran  f ) ) )  ->  ( N `  ( N `  U. ran  f ) )  =  ( N `  U. ran  f ) )
2820, 27sseqtrd 3386 . . . . . . . 8  |-  ( (
ph  /\  ( f : T --> ( ~P S  i^i  Fin )  /\  T  C_  ( N `  U. ran  f ) ) )  ->  ( N `  T )  C_  ( N `  U. ran  f
) )
2917, 28eqsstrd 3384 . . . . . . 7  |-  ( (
ph  /\  ( f : T --> ( ~P S  i^i  Fin )  /\  T  C_  ( N `  U. ran  f ) ) )  ->  ( N `  S )  C_  ( N `  U. ran  f
) )
3016, 29sstrd 3360 . . . . . 6  |-  ( (
ph  /\  ( f : T --> ( ~P S  i^i  Fin )  /\  T  C_  ( N `  U. ran  f ) ) )  ->  S  C_  ( N `  U. ran  f
) )
3113, 2, 3, 30, 25, 14mrissmrcd 13867 . . . . 5  |-  ( (
ph  /\  ( f : T --> ( ~P S  i^i  Fin )  /\  T  C_  ( N `  U. ran  f ) ) )  ->  S  =  U. ran  f )
3212, 31jca 520 . . . 4  |-  ( (
ph  /\  ( f : T --> ( ~P S  i^i  Fin )  /\  T  C_  ( N `  U. ran  f ) ) )  ->  ( f : T --> ( ~P S  i^i  Fin )  /\  S  =  U. ran  f ) )
3332ex 425 . . 3  |-  ( ph  ->  ( ( f : T --> ( ~P S  i^i  Fin )  /\  T  C_  ( N `  U. ran  f ) )  -> 
( f : T --> ( ~P S  i^i  Fin )  /\  S  =  U. ran  f ) ) )
3433eximdv 1633 . 2  |-  ( ph  ->  ( E. f ( f : T --> ( ~P S  i^i  Fin )  /\  T  C_  ( N `
 U. ran  f
) )  ->  E. f
( f : T --> ( ~P S  i^i  Fin )  /\  S  =  U. ran  f ) ) )
3511, 34mpd 15 1  |-  ( ph  ->  E. f ( f : T --> ( ~P S  i^i  Fin )  /\  S  =  U. ran  f ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360   E.wex 1551    = wceq 1653    e. wcel 1726    i^i cin 3321    C_ wss 3322   ~Pcpw 3801   U.cuni 4017   ran crn 4881   -->wf 5452   ` cfv 5456   Fincfn 7111  Moorecmre 13809  mrClscmrc 13810  mrIndcmri 13811  ACScacs 13812
This theorem is referenced by:  acsinfd  14608  acsdomd  14609
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-reg 7562  ax-inf2 7598  ax-ac2 8345  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-iin 4098  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-se 4544  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-isom 5465  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-1o 6726  df-oadd 6730  df-er 6907  df-en 7112  df-dom 7113  df-sdom 7114  df-fin 7115  df-r1 7692  df-rank 7693  df-card 7828  df-ac 7999  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-nn 10003  df-2 10060  df-3 10061  df-4 10062  df-5 10063  df-6 10064  df-7 10065  df-8 10066  df-9 10067  df-10 10068  df-n0 10224  df-z 10285  df-dec 10385  df-uz 10491  df-fz 11046  df-struct 13473  df-ndx 13474  df-slot 13475  df-base 13476  df-tset 13550  df-ple 13551  df-ocomp 13552  df-mre 13813  df-mrc 13814  df-mri 13815  df-acs 13816  df-preset 14387  df-drs 14388  df-poset 14405  df-ipo 14580
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