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Theorem ismkvmap 7145
Description: The predicate of being Markov stated in terms of set exponentiation. (Contributed by Jim Kingdon, 18-Mar-2023.)
Assertion
Ref Expression
ismkvmap  |-  ( A  e.  V  ->  ( A  e. Markov  <->  A. f  e.  ( 2o  ^m  A ) ( -.  A. x  e.  A  ( f `  x )  =  1o 
->  E. x  e.  A  ( f `  x
)  =  (/) ) ) )
Distinct variable groups:    A, f, x   
f, V
Allowed substitution hint:    V( x)

Proof of Theorem ismkvmap
StepHypRef Expression
1 ismkv 7144 . . 3  |-  ( A  e.  V  ->  ( A  e. Markov  <->  A. f ( f : A --> 2o  ->  ( -.  A. x  e.  A  ( f `  x )  =  1o 
->  E. x  e.  A  ( f `  x
)  =  (/) ) ) ) )
2 2onn 6515 . . . . . 6  |-  2o  e.  om
3 elmapg 6654 . . . . . 6  |-  ( ( 2o  e.  om  /\  A  e.  V )  ->  ( f  e.  ( 2o  ^m  A )  <-> 
f : A --> 2o ) )
42, 3mpan 424 . . . . 5  |-  ( A  e.  V  ->  (
f  e.  ( 2o 
^m  A )  <->  f : A
--> 2o ) )
54imbi1d 231 . . . 4  |-  ( A  e.  V  ->  (
( f  e.  ( 2o  ^m  A )  ->  ( -.  A. x  e.  A  (
f `  x )  =  1o  ->  E. x  e.  A  ( f `  x )  =  (/) ) )  <->  ( f : A --> 2o  ->  ( -.  A. x  e.  A  ( f `  x
)  =  1o  ->  E. x  e.  A  ( f `  x )  =  (/) ) ) ) )
65albidv 1824 . . 3  |-  ( A  e.  V  ->  ( A. f ( f  e.  ( 2o  ^m  A
)  ->  ( -.  A. x  e.  A  ( f `  x )  =  1o  ->  E. x  e.  A  ( f `  x )  =  (/) ) )  <->  A. f
( f : A --> 2o  ->  ( -.  A. x  e.  A  (
f `  x )  =  1o  ->  E. x  e.  A  ( f `  x )  =  (/) ) ) ) )
71, 6bitr4d 191 . 2  |-  ( A  e.  V  ->  ( A  e. Markov  <->  A. f ( f  e.  ( 2o  ^m  A )  ->  ( -.  A. x  e.  A  ( f `  x
)  =  1o  ->  E. x  e.  A  ( f `  x )  =  (/) ) ) ) )
8 df-ral 2460 . 2  |-  ( A. f  e.  ( 2o  ^m  A ) ( -. 
A. x  e.  A  ( f `  x
)  =  1o  ->  E. x  e.  A  ( f `  x )  =  (/) )  <->  A. f
( f  e.  ( 2o  ^m  A )  ->  ( -.  A. x  e.  A  (
f `  x )  =  1o  ->  E. x  e.  A  ( f `  x )  =  (/) ) ) )
97, 8bitr4di 198 1  |-  ( A  e.  V  ->  ( A  e. Markov  <->  A. f  e.  ( 2o  ^m  A ) ( -.  A. x  e.  A  ( f `  x )  =  1o 
->  E. x  e.  A  ( f `  x
)  =  (/) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 105   A.wal 1351    = wceq 1353    e. wcel 2148   A.wral 2455   E.wrex 2456   (/)c0 3422   omcom 4585   -->wf 5207   ` cfv 5211  (class class class)co 5868   1oc1o 6403   2oc2o 6404    ^m cmap 6641  Markovcmarkov 7142
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-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4205  ax-un 4429  ax-setind 4532
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-br 4001  df-opab 4062  df-id 4289  df-suc 4367  df-iom 4586  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-iota 5173  df-fun 5213  df-fn 5214  df-f 5215  df-fv 5219  df-ov 5871  df-oprab 5872  df-mpo 5873  df-1o 6410  df-2o 6411  df-map 6643  df-markov 7143
This theorem is referenced by:  ismkvnex  7146  fodjumkvlemres  7150  enmkvlem  7152  ismkvnnlem  14423
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