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Theorem isacs 13881
 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 5761 . 2 ACS
2 elfvex 5761 . . 3 Moore
4 fveq2 5731 . . . . . 6 Moore Moore
5 pweq 3804 . . . . . . . . 9
65, 5feq23d 5591 . . . . . . . 8
75raleqdv 2912 . . . . . . . 8
86, 7anbi12d 693 . . . . . . 7
98exbidv 1637 . . . . . 6
104, 9rabeqbidv 2953 . . . . 5 Moore Moore
11 df-acs 13819 . . . . 5 ACS Moore
12 fvex 5745 . . . . . 6 Moore
1312rabex 4357 . . . . 5 Moore
1410, 11, 13fvmpt 5809 . . . 4 ACS Moore
1514eleq2d 2505 . . 3 ACS Moore
16 eleq2 2499 . . . . . . . 8
1716bibi1d 312 . . . . . . 7
1817ralbidv 2727 . . . . . 6
1918anbi2d 686 . . . . 5
2019exbidv 1637 . . . 4
2120elrab 3094 . . 3 Moore Moore
2215, 21syl6bb 254 . 2 ACS Moore
231, 3, 22pm5.21nii 344 1 ACS Moore
 Colors of variables: wff set class Syntax hints:   wb 178   wa 360  wex 1551   wceq 1653   wcel 1726  wral 2707  crab 2711  cvv 2958   cin 3321   wss 3322  cpw 3801  cuni 4017  cima 4884  wf 5453  cfv 5457  cfn 7112  Moorecmre 13812  ACScacs 13815 This theorem is referenced by:  acsmre  13882  isacs2  13883  isacs1i  13887  mreacs  13888 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  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-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-fv 5465  df-acs 13819
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