Theorem ismkv 7117

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 5321	. . . 4 |- ( y = A -> ( f : y --> 2o <-> f : A --> 2o ) )
2		raleq 2661	. . . . . 6 |- ( y = A -> ( A. x e. y ( f ` x ) = 1o <-> A. x e. A ( f ` x ) = 1o ) )
3	2	notbid 657	. . . . 5 |- ( y = A -> ( -. A. x e. y ( f ` x ) = 1o <-> -. A. x e. A ( f ` x ) = 1o ) )
4		rexeq 2662	. . . . 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 1812	. 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 7116	. 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 2873	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 1341   = wceq 1343   e. wcel 2136   A.wral 2444   E.wrex 2445   (/)c0 3409   -->wf 5184   ` cfv 5188   1oc1o 6377   2oc2o 6378  Markovcmarkov 7115

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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-fn 5191  df-f 5192  df-markov 7116

This theorem is referenced by:  ismkvmap  7118  omnimkv  7120  mkvprop  7122  omniwomnimkv  7131