Theorem ismkv 7145

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 5345	. . . 4 |- ( y = A -> ( f : y --> 2o <-> f : A --> 2o ) )
2		raleq 2672	. . . . . 6 |- ( y = A -> ( A. x e. y ( f ` x ) = 1o <-> A. x e. A ( f ` x ) = 1o ) )
3	2	notbid 667	. . . . 5 |- ( y = A -> ( -. A. x e. y ( f ` x ) = 1o <-> -. A. x e. A ( f ` x ) = 1o ) )
4		rexeq 2673	. . . . 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 1824	. 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 7144	. 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 2884	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 1351   = wceq 1353   e. wcel 2148   A.wral 2455   E.wrex 2456   (/)c0 3422   -->wf 5208   ` cfv 5212   1oc1o 6404   2oc2o 6405  Markovcmarkov 7143

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-fn 5215  df-f 5216  df-markov 7144

This theorem is referenced by:  ismkvmap  7146  omnimkv  7148  mkvprop  7150  omniwomnimkv  7159