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Theorem acsmap2d 16376
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 16375 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 15497,  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 15513 . . . 4  |-  ( ph  ->  A  e.  (Moore `  X ) )
5 acsmap2d.4 . . . 4  |-  ( ph  ->  S  e.  I )
63, 4, 5mrissd 15493 . . 3  |-  ( ph  ->  S  C_  X )
7 acsmap2d.5 . . . . 5  |-  ( ph  ->  T  C_  X )
84, 2, 7mrcssidd 15482 . . . 4  |-  ( ph  ->  T  C_  ( N `  T ) )
9 acsmap2d.6 . . . 4  |-  ( ph  ->  ( N `  S
)  =  ( N `
 T ) )
108, 9sseqtr4d 3507 . . 3  |-  ( ph  ->  T  C_  ( N `  S ) )
111, 2, 6, 10acsmapd 16375 . 2  |-  ( ph  ->  E. f ( f : T --> ( ~P S  i^i  Fin )  /\  T  C_  ( N `
 U. ran  f
) ) )
12 simprl 762 . . . . 5  |-  ( (
ph  /\  ( f : T --> ( ~P S  i^i  Fin )  /\  T  C_  ( N `  U. ran  f ) ) )  ->  f : T --> ( ~P S  i^i  Fin ) )
134adantr 466 . . . . . 6  |-  ( (
ph  /\  ( f : T --> ( ~P S  i^i  Fin )  /\  T  C_  ( N `  U. ran  f ) ) )  ->  A  e.  (Moore `  X ) )
145adantr 466 . . . . . . . . 9  |-  ( (
ph  /\  ( f : T --> ( ~P S  i^i  Fin )  /\  T  C_  ( N `  U. ran  f ) ) )  ->  S  e.  I
)
153, 13, 14mrissd 15493 . . . . . . . 8  |-  ( (
ph  /\  ( f : T --> ( ~P S  i^i  Fin )  /\  T  C_  ( N `  U. ran  f ) ) )  ->  S  C_  X
)
1613, 2, 15mrcssidd 15482 . . . . . . 7  |-  ( (
ph  /\  ( f : T --> ( ~P S  i^i  Fin )  /\  T  C_  ( N `  U. ran  f ) ) )  ->  S  C_  ( N `  S )
)
179adantr 466 . . . . . . . 8  |-  ( (
ph  /\  ( f : T --> ( ~P S  i^i  Fin )  /\  T  C_  ( N `  U. ran  f ) ) )  ->  ( N `  S )  =  ( N `  T ) )
18 simprr 764 . . . . . . . . . 10  |-  ( (
ph  /\  ( f : T --> ( ~P S  i^i  Fin )  /\  T  C_  ( N `  U. ran  f ) ) )  ->  T  C_  ( N `  U. ran  f
) )
1913, 2mrcssvd 15480 . . . . . . . . . 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 15481 . . . . . . . . 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 5752 . . . . . . . . . . . . . 14  |-  ( f : T --> ( ~P S  i^i  Fin )  ->  ran  f  C_  ( ~P S  i^i  Fin )
)
2221unissd 4246 . . . . . . . . . . . . 13  |-  ( f : T --> ( ~P S  i^i  Fin )  ->  U. ran  f  C_  U. ( ~P S  i^i  Fin ) )
23 unifpw 7883 . . . . . . . . . . . . 13  |-  U. ( ~P S  i^i  Fin )  =  S
2422, 23syl6sseq 3516 . . . . . . . . . . . 12  |-  ( f : T --> ( ~P S  i^i  Fin )  ->  U. ran  f  C_  S )
2524ad2antrl 732 . . . . . . . . . . 11  |-  ( (
ph  /\  ( f : T --> ( ~P S  i^i  Fin )  /\  T  C_  ( N `  U. ran  f ) ) )  ->  U. ran  f  C_  S )
2625, 15sstrd 3480 . . . . . . . . . 10  |-  ( (
ph  /\  ( f : T --> ( ~P S  i^i  Fin )  /\  T  C_  ( N `  U. ran  f ) ) )  ->  U. ran  f  C_  X )
2713, 2, 26mrcidmd 15483 . . . . . . . . 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 3506 . . . . . . . 8  |-  ( (
ph  /\  ( f : T --> ( ~P S  i^i  Fin )  /\  T  C_  ( N `  U. ran  f ) ) )  ->  ( N `  T )  C_  ( N `  U. ran  f
) )
2917, 28eqsstrd 3504 . . . . . . 7  |-  ( (
ph  /\  ( f : T --> ( ~P S  i^i  Fin )  /\  T  C_  ( N `  U. ran  f ) ) )  ->  ( N `  S )  C_  ( N `  U. ran  f
) )
3016, 29sstrd 3480 . . . . . 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 15497 . . . . 5  |-  ( (
ph  /\  ( f : T --> ( ~P S  i^i  Fin )  /\  T  C_  ( N `  U. ran  f ) ) )  ->  S  =  U. ran  f )
3212, 31jca 534 . . . 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 435 . . 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 1757 . 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 setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437   E.wex 1659    e. wcel 1870    i^i cin 3441    C_ wss 3442   ~Pcpw 3985   U.cuni 4222   ran crn 4855   -->wf 5597   ` cfv 5601   Fincfn 7577  Moorecmre 15439  mrClscmrc 15440  mrIndcmri 15441  ACScacs 15442
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-reg 8107  ax-inf2 8146  ax-ac2 8891  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-iin 4305  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-se 4814  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-isom 5610  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-oadd 7194  df-er 7371  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-r1 8234  df-rank 8235  df-card 8372  df-ac 8545  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-uz 11160  df-fz 11783  df-struct 15086  df-ndx 15087  df-slot 15088  df-base 15089  df-tset 15171  df-ple 15172  df-ocomp 15173  df-mre 15443  df-mrc 15444  df-mri 15445  df-acs 15446  df-preset 16124  df-drs 16125  df-poset 16142  df-ipo 16349
This theorem is referenced by:  acsinfd  16377  acsdomd  16378
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