Theorem ismkv 7219

Description: The predicate of being Markov. (Contributed by Jim Kingdon, 18-Mar-2023.)

Assertion

Ref	Expression
ismkv	|- ( A e. V -> ( A e. Markov <-> A. f ( f : A --> 2o -> ( -. A. x e. A ( f ` x ) = 1o -> E. x e. A ( f ` x ) = (/) ) ) ) )

Distinct variable group:   A, f, x

Allowed substitution hints:   V( x, f)

Proof of Theorem ismkv

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		feq2 5391	. . . 4 |- ( y = A -> ( f : y --> 2o <-> f : A --> 2o ) )
2		raleq 2693	. . . . . 6 |- ( y = A -> ( A. x e. y ( f ` x ) = 1o <-> A. x e. A ( f ` x ) = 1o ) )
3	2	notbid 668	. . . . 5 |- ( y = A -> ( -. A. x e. y ( f ` x ) = 1o <-> -. A. x e. A ( f ` x ) = 1o ) )
4		rexeq 2694	. . . . 5 |- ( y = A -> ( E. x e. y ( f ` x ) = (/) <-> E. x e. A ( f ` x ) = (/) ) )
5	3, 4	imbi12d 234	. . . 4 |- ( y = A -> ( ( -. A. x e. y ( f ` x ) = 1o -> E. x e. y ( f ` x ) = (/) ) <-> ( -. A. x e. A ( f ` x ) = 1o -> E. x e. A ( f ` x ) = (/) ) ) )
6	1, 5	imbi12d 234	. . 3 |- ( y = A -> ( ( f : y --> 2o -> ( -. A. x e. y ( f ` x ) = 1o -> E. x e. y ( f ` x ) = (/) ) ) <-> ( f : A --> 2o -> ( -. A. x e. A ( f ` x ) = 1o -> E. x e. A ( f ` x ) = (/) ) ) ) )
7	6	albidv 1838	. 2 |- ( y = A -> ( A. f ( f : y --> 2o -> ( -. A. x e. y ( f ` x ) = 1o -> E. x e. y ( f ` x ) = (/) ) ) <-> A. f ( f : A --> 2o -> ( -. A. x e. A ( f ` x ) = 1o -> E. x e. A ( f ` x ) = (/) ) ) ) )
8		df-markov 7218	. 2 |- Markov = { y | A. f ( f : y --> 2o -> ( -. A. x e. y ( f ` x ) = 1o -> E. x e. y ( f ` x ) = (/) ) ) }
9	7, 8	elab2g 2911	1 |- ( A e. V -> ( A e. Markov <-> A. f ( f : A --> 2o -> ( -. A. x e. A ( f ` x ) = 1o -> E. x e. A ( f ` x ) = (/) ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 105   A.wal 1362   = wceq 1364   e. wcel 2167   A.wral 2475   E.wrex 2476   (/)c0 3450   -->wf 5254   ` cfv 5258   1oc1o 6467   2oc2o 6468  Markovcmarkov 7217

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-fn 5261  df-f 5262  df-markov 7218

This theorem is referenced by:  ismkvmap  7220  omnimkv  7222  mkvprop  7224  omniwomnimkv  7233