Metamath Proof Explorer

Theorem isnacs

Description: Expand definition of Noetherian-type closure system. (Contributed by Stefan O'Rear, 4-Apr-2015)

Ref Expression
Hypothesis isnacs.f 
|- F = ( mrCls ` C )
Assertion isnacs 
|- ( C e. ( NoeACS ` X ) <-> ( C e. ( ACS ` X ) /\ A. s e. C E. g e. ( ~P X i^i Fin ) s = ( F ` g ) ) )

Proof

Step Hyp Ref Expression
1 isnacs.f 
 |-  F = ( mrCls ` C )
2 elfvex 
 |-  ( C e. ( NoeACS ` X ) -> X e. _V )
3 elfvex 
 |-  ( C e. ( ACS ` X ) -> X e. _V )
4 3 adantr 
 |-  ( ( C e. ( ACS ` X ) /\ A. s e. C E. g e. ( ~P X i^i Fin ) s = ( F ` g ) ) -> X e. _V )
5 fveq2 
 |-  ( x = X -> ( ACS ` x ) = ( ACS ` X ) )
6 pweq 
 |-  ( x = X -> ~P x = ~P X )
7 6 ineq1d 
 |-  ( x = X -> ( ~P x i^i Fin ) = ( ~P X i^i Fin ) )
8 7 rexeqdv 
 |-  ( x = X -> ( E. g e. ( ~P x i^i Fin ) s = ( ( mrCls ` c ) ` g ) <-> E. g e. ( ~P X i^i Fin ) s = ( ( mrCls ` c ) ` g ) ) )
9 8 ralbidv 
 |-  ( x = X -> ( A. s e. c E. g e. ( ~P x i^i Fin ) s = ( ( mrCls ` c ) ` g ) <-> A. s e. c E. g e. ( ~P X i^i Fin ) s = ( ( mrCls ` c ) ` g ) ) )
10 5 9 rabeqbidv 
 |-  ( x = X -> { c e. ( ACS ` x ) | A. s e. c E. g e. ( ~P x i^i Fin ) s = ( ( mrCls ` c ) ` g ) } = { c e. ( ACS ` X ) | A. s e. c E. g e. ( ~P X i^i Fin ) s = ( ( mrCls ` c ) ` g ) } )
11 df-nacs 
 |-  NoeACS = ( x e. _V |-> { c e. ( ACS ` x ) | A. s e. c E. g e. ( ~P x i^i Fin ) s = ( ( mrCls ` c ) ` g ) } )
12 fvex 
 |-  ( ACS ` X ) e. _V
13 12 rabex 
 |-  { c e. ( ACS ` X ) | A. s e. c E. g e. ( ~P X i^i Fin ) s = ( ( mrCls ` c ) ` g ) } e. _V
14 10 11 13 fvmpt 
 |-  ( X e. _V -> ( NoeACS ` X ) = { c e. ( ACS ` X ) | A. s e. c E. g e. ( ~P X i^i Fin ) s = ( ( mrCls ` c ) ` g ) } )
15 14 eleq2d 
 |-  ( X e. _V -> ( C e. ( NoeACS ` X ) <-> C e. { c e. ( ACS ` X ) | A. s e. c E. g e. ( ~P X i^i Fin ) s = ( ( mrCls ` c ) ` g ) } ) )
16 fveq2 
 |-  ( c = C -> ( mrCls ` c ) = ( mrCls ` C ) )
17 16 1 eqtr4di 
 |-  ( c = C -> ( mrCls ` c ) = F )
18 17 fveq1d 
 |-  ( c = C -> ( ( mrCls ` c ) ` g ) = ( F ` g ) )
19 18 eqeq2d 
 |-  ( c = C -> ( s = ( ( mrCls ` c ) ` g ) <-> s = ( F ` g ) ) )
20 19 rexbidv 
 |-  ( c = C -> ( E. g e. ( ~P X i^i Fin ) s = ( ( mrCls ` c ) ` g ) <-> E. g e. ( ~P X i^i Fin ) s = ( F ` g ) ) )
21 20 raleqbi1dv 
 |-  ( c = C -> ( A. s e. c E. g e. ( ~P X i^i Fin ) s = ( ( mrCls ` c ) ` g ) <-> A. s e. C E. g e. ( ~P X i^i Fin ) s = ( F ` g ) ) )
22 21 elrab 
 |-  ( C e. { c e. ( ACS ` X ) | A. s e. c E. g e. ( ~P X i^i Fin ) s = ( ( mrCls ` c ) ` g ) } <-> ( C e. ( ACS ` X ) /\ A. s e. C E. g e. ( ~P X i^i Fin ) s = ( F ` g ) ) )
23 15 22 bitrdi 
 |-  ( X e. _V -> ( C e. ( NoeACS ` X ) <-> ( C e. ( ACS ` X ) /\ A. s e. C E. g e. ( ~P X i^i Fin ) s = ( F ` g ) ) ) )
24 2 4 23 pm5.21nii 
 |-  ( C e. ( NoeACS ` X ) <-> ( C e. ( ACS ` X ) /\ A. s e. C E. g e. ( ~P X i^i Fin ) s = ( F ` g ) ) )