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Theorem isacs 14909
 Description: A set is an algebraic closure system iff it is specified by some function of the finite subsets, such that a set is closed iff it does not expand under the operation. (Contributed by Stefan O'Rear, 2-Apr-2015.)
Assertion
Ref Expression
isacs ACS Moore
Distinct variable groups:   ,,   ,,

Proof of Theorem isacs
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elfvex 5893 . 2 ACS
2 elfvex 5893 . . 3 Moore
4 fveq2 5866 . . . . . 6 Moore Moore
5 pweq 4013 . . . . . . . . 9
65, 5feq23d 5726 . . . . . . . 8
75raleqdv 3064 . . . . . . . 8
86, 7anbi12d 710 . . . . . . 7
98exbidv 1690 . . . . . 6
104, 9rabeqbidv 3108 . . . . 5 Moore Moore
11 df-acs 14847 . . . . 5 ACS Moore
12 fvex 5876 . . . . . 6 Moore
1312rabex 4598 . . . . 5 Moore
1410, 11, 13fvmpt 5951 . . . 4 ACS Moore
1514eleq2d 2537 . . 3 ACS Moore
16 eleq2 2540 . . . . . . . 8
1716bibi1d 319 . . . . . . 7
1817ralbidv 2903 . . . . . 6
1918anbi2d 703 . . . . 5
2019exbidv 1690 . . . 4
2120elrab 3261 . . 3 Moore Moore
2215, 21syl6bb 261 . 2 ACS Moore
231, 3, 22pm5.21nii 353 1 ACS Moore
 Colors of variables: wff setvar class Syntax hints:   wb 184   wa 369   wceq 1379  wex 1596   wcel 1767  wral 2814  crab 2818  cvv 3113   cin 3475   wss 3476  cpw 4010  cuni 4245  cima 5002  wf 5584  cfv 5588  cfn 7517  Moorecmre 14840  ACScacs 14843 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-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  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-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-fv 5596  df-acs 14847 This theorem is referenced by:  acsmre  14910  isacs2  14911  isacs1i  14915  mreacs  14916
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