Theorem ismkv 7027

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 5256	. . . 4 |- ( y = A -> ( f : y --> 2o <-> f : A --> 2o ) )
2		raleq 2626	. . . . . 6 |- ( y = A -> ( A. x e. y ( f ` x ) = 1o <-> A. x e. A ( f ` x ) = 1o ) )
3	2	notbid 656	. . . . 5 |- ( y = A -> ( -. A. x e. y ( f ` x ) = 1o <-> -. A. x e. A ( f ` x ) = 1o ) )
4		rexeq 2627	. . . . 5 |- ( y = A -> ( E. x e. y ( f ` x ) = (/) <-> E. x e. A ( f ` x ) = (/) ) )
5	3, 4	imbi12d 233	. . . 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 233	. . 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 1796	. 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 7026	. 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 2831	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 104   A.wal 1329   = wceq 1331   e. wcel 1480   A.wral 2416   E.wrex 2417   (/)c0 3363   -->wf 5119   ` cfv 5123   1oc1o 6306   2oc2o 6307  Markovcmarkov 7025

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-fn 5126  df-f 5127  df-markov 7026

This theorem is referenced by:  ismkvmap  7028  omnimkv  7030  mkvprop  7032