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Metamath Proof Explorer

Theorem axdc2

Description: An apparent strengthening of ax-dc (but derived from it) which shows that there is a denumerable sequence g for any function that maps elements of a set A to nonempty subsets of A such that g ( x + 1 ) e. F ( g ( x ) ) for all x e. _om . The finitistic version of this can be proven by induction, but the infinite version requires this new axiom. (Contributed by Mario Carneiro, 25-Jan-2013)

Ref Expression
Hypothesis axdc2.1 
|- A e. _V
Assertion axdc2 
|- ( ( A =/= (/) /\ F : A --> ( ~P A \ { (/) } ) ) -> E. g ( g : _om --> A /\ A. k e. _om ( g ` suc k ) e. ( F ` ( g ` k ) ) ) )

Proof

Step Hyp Ref Expression
1 axdc2.1 
 |-  A e. _V
2 eleq1w 
 |-  ( s = x -> ( s e. A <-> x e. A ) )
3 2 adantr 
 |-  ( ( s = x /\ t = y ) -> ( s e. A <-> x e. A ) )
4 fveq2 
 |-  ( s = x -> ( F ` s ) = ( F ` x ) )
5 4 eleq2d 
 |-  ( s = x -> ( t e. ( F ` s ) <-> t e. ( F ` x ) ) )
6 eleq1w 
 |-  ( t = y -> ( t e. ( F ` x ) <-> y e. ( F ` x ) ) )
7 5 6 sylan9bb 
 |-  ( ( s = x /\ t = y ) -> ( t e. ( F ` s ) <-> y e. ( F ` x ) ) )
8 3 7 anbi12d 
 |-  ( ( s = x /\ t = y ) -> ( ( s e. A /\ t e. ( F ` s ) ) <-> ( x e. A /\ y e. ( F ` x ) ) ) )
9 8 cbvopabv 
 |-  { <. s , t >. | ( s e. A /\ t e. ( F ` s ) ) } = { <. x , y >. | ( x e. A /\ y e. ( F ` x ) ) }
10 fveq2 
 |-  ( n = x -> ( h ` n ) = ( h ` x ) )
11 10 cbvmptv 
 |-  ( n e. _om |-> ( h ` n ) ) = ( x e. _om |-> ( h ` x ) )
12 1 9 11 axdc2lem 
 |-  ( ( A =/= (/) /\ F : A --> ( ~P A \ { (/) } ) ) -> E. g ( g : _om --> A /\ A. k e. _om ( g ` suc k ) e. ( F ` ( g ` k ) ) ) )