MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  acsmap2d Structured version   Unicode version

Theorem acsmap2d 15683
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 15682 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 14912,  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 14928 . . . 4  |-  ( ph  ->  A  e.  (Moore `  X ) )
5 acsmap2d.4 . . . 4  |-  ( ph  ->  S  e.  I )
63, 4, 5mrissd 14908 . . 3  |-  ( ph  ->  S  C_  X )
7 acsmap2d.5 . . . . 5  |-  ( ph  ->  T  C_  X )
84, 2, 7mrcssidd 14897 . . . 4  |-  ( ph  ->  T  C_  ( N `  T ) )
9 acsmap2d.6 . . . 4  |-  ( ph  ->  ( N `  S
)  =  ( N `
 T ) )
108, 9sseqtr4d 3546 . . 3  |-  ( ph  ->  T  C_  ( N `  S ) )
111, 2, 6, 10acsmapd 15682 . 2  |-  ( ph  ->  E. f ( f : T --> ( ~P S  i^i  Fin )  /\  T  C_  ( N `
 U. ran  f
) ) )
12 simprl 755 . . . . 5  |-  ( (
ph  /\  ( f : T --> ( ~P S  i^i  Fin )  /\  T  C_  ( N `  U. ran  f ) ) )  ->  f : T --> ( ~P S  i^i  Fin ) )
134adantr 465 . . . . . 6  |-  ( (
ph  /\  ( f : T --> ( ~P S  i^i  Fin )  /\  T  C_  ( N `  U. ran  f ) ) )  ->  A  e.  (Moore `  X ) )
145adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  ( f : T --> ( ~P S  i^i  Fin )  /\  T  C_  ( N `  U. ran  f ) ) )  ->  S  e.  I
)
153, 13, 14mrissd 14908 . . . . . . . 8  |-  ( (
ph  /\  ( f : T --> ( ~P S  i^i  Fin )  /\  T  C_  ( N `  U. ran  f ) ) )  ->  S  C_  X
)
1613, 2, 15mrcssidd 14897 . . . . . . 7  |-  ( (
ph  /\  ( f : T --> ( ~P S  i^i  Fin )  /\  T  C_  ( N `  U. ran  f ) ) )  ->  S  C_  ( N `  S )
)
179adantr 465 . . . . . . . 8  |-  ( (
ph  /\  ( f : T --> ( ~P S  i^i  Fin )  /\  T  C_  ( N `  U. ran  f ) ) )  ->  ( N `  S )  =  ( N `  T ) )
18 simprr 756 . . . . . . . . . 10  |-  ( (
ph  /\  ( f : T --> ( ~P S  i^i  Fin )  /\  T  C_  ( N `  U. ran  f ) ) )  ->  T  C_  ( N `  U. ran  f
) )
1913, 2mrcssvd 14895 . . . . . . . . . 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 14896 . . . . . . . . 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 5743 . . . . . . . . . . . . . 14  |-  ( f : T --> ( ~P S  i^i  Fin )  ->  ran  f  C_  ( ~P S  i^i  Fin )
)
2221unissd 4275 . . . . . . . . . . . . 13  |-  ( f : T --> ( ~P S  i^i  Fin )  ->  U. ran  f  C_  U. ( ~P S  i^i  Fin ) )
23 unifpw 7835 . . . . . . . . . . . . 13  |-  U. ( ~P S  i^i  Fin )  =  S
2422, 23syl6sseq 3555 . . . . . . . . . . . 12  |-  ( f : T --> ( ~P S  i^i  Fin )  ->  U. ran  f  C_  S )
2524ad2antrl 727 . . . . . . . . . . 11  |-  ( (
ph  /\  ( f : T --> ( ~P S  i^i  Fin )  /\  T  C_  ( N `  U. ran  f ) ) )  ->  U. ran  f  C_  S )
2625, 15sstrd 3519 . . . . . . . . . 10  |-  ( (
ph  /\  ( f : T --> ( ~P S  i^i  Fin )  /\  T  C_  ( N `  U. ran  f ) ) )  ->  U. ran  f  C_  X )
2713, 2, 26mrcidmd 14898 . . . . . . . . 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 3545 . . . . . . . 8  |-  ( (
ph  /\  ( f : T --> ( ~P S  i^i  Fin )  /\  T  C_  ( N `  U. ran  f ) ) )  ->  ( N `  T )  C_  ( N `  U. ran  f
) )
2917, 28eqsstrd 3543 . . . . . . 7  |-  ( (
ph  /\  ( f : T --> ( ~P S  i^i  Fin )  /\  T  C_  ( N `  U. ran  f ) ) )  ->  ( N `  S )  C_  ( N `  U. ran  f
) )
3016, 29sstrd 3519 . . . . . 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 14912 . . . . 5  |-  ( (
ph  /\  ( f : T --> ( ~P S  i^i  Fin )  /\  T  C_  ( N `  U. ran  f ) ) )  ->  S  =  U. ran  f )
3212, 31jca 532 . . . 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 434 . . 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 1686 . 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 369    = wceq 1379   E.wex 1596    e. wcel 1767    i^i cin 3480    C_ wss 3481   ~Pcpw 4016   U.cuni 4251   ran crn 5006   -->wf 5590   ` cfv 5594   Fincfn 7528  Moorecmre 14854  mrClscmrc 14855  mrIndcmri 14856  ACScacs 14857
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-reg 8030  ax-inf2 8070  ax-ac2 8855  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-iin 4334  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-se 4845  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-r1 8194  df-rank 8195  df-card 8332  df-ac 8509  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-2 10606  df-3 10607  df-4 10608  df-5 10609  df-6 10610  df-7 10611  df-8 10612  df-9 10613  df-10 10614  df-n0 10808  df-z 10877  df-dec 10989  df-uz 11095  df-fz 11685  df-struct 14509  df-ndx 14510  df-slot 14511  df-base 14512  df-tset 14591  df-ple 14592  df-ocomp 14593  df-mre 14858  df-mrc 14859  df-mri 14860  df-acs 14861  df-preset 15432  df-drs 15433  df-poset 15450  df-ipo 15656
This theorem is referenced by:  acsinfd  15684  acsdomd  15685
  Copyright terms: Public domain W3C validator