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Theorem fodjumkvlemres 7150
Description: Lemma for fodjumkv 7151. The final result with  P expressed as a local definition. (Contributed by Jim Kingdon, 25-Mar-2023.)
Hypotheses
Ref Expression
fodjumkv.o  |-  ( ph  ->  M  e. Markov )
fodjumkv.fo  |-  ( ph  ->  F : M -onto-> ( A B ) )
fodjumkv.p  |-  P  =  ( y  e.  M  |->  if ( E. z  e.  A  ( F `  y )  =  (inl
`  z ) ,  (/) ,  1o ) )
Assertion
Ref Expression
fodjumkvlemres  |-  ( ph  ->  ( A  =/=  (/)  ->  E. x  x  e.  A )
)
Distinct variable groups:    ph, y, z   
y, M, z    z, A    z, B    z, F    x, A, z    y, A   
y, F    y, P, z
Allowed substitution hints:    ph( x)    B( x, y)    P( x)    F( x)    M( x)

Proof of Theorem fodjumkvlemres
Dummy variables  v  f  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fodjumkv.fo . . . . . 6  |-  ( ph  ->  F : M -onto-> ( A B ) )
21adantr 276 . . . . 5  |-  ( (
ph  /\  A. w  e.  M  ( P `  w )  =  1o )  ->  F : M -onto-> ( A B ) )
3 fodjumkv.p . . . . 5  |-  P  =  ( y  e.  M  |->  if ( E. z  e.  A  ( F `  y )  =  (inl
`  z ) ,  (/) ,  1o ) )
4 simpr 110 . . . . 5  |-  ( (
ph  /\  A. w  e.  M  ( P `  w )  =  1o )  ->  A. w  e.  M  ( P `  w )  =  1o )
52, 3, 4fodju0 7138 . . . 4  |-  ( (
ph  /\  A. w  e.  M  ( P `  w )  =  1o )  ->  A  =  (/) )
65ex 115 . . 3  |-  ( ph  ->  ( A. w  e.  M  ( P `  w )  =  1o 
->  A  =  (/) ) )
76necon3ad 2389 . 2  |-  ( ph  ->  ( A  =/=  (/)  ->  -.  A. w  e.  M  ( P `  w )  =  1o ) )
8 fveq1 5509 . . . . . . 7  |-  ( f  =  P  ->  (
f `  w )  =  ( P `  w ) )
98eqeq1d 2186 . . . . . 6  |-  ( f  =  P  ->  (
( f `  w
)  =  1o  <->  ( P `  w )  =  1o ) )
109ralbidv 2477 . . . . 5  |-  ( f  =  P  ->  ( A. w  e.  M  ( f `  w
)  =  1o  <->  A. w  e.  M  ( P `  w )  =  1o ) )
1110notbid 667 . . . 4  |-  ( f  =  P  ->  ( -.  A. w  e.  M  ( f `  w
)  =  1o  <->  -.  A. w  e.  M  ( P `  w )  =  1o ) )
128eqeq1d 2186 . . . . 5  |-  ( f  =  P  ->  (
( f `  w
)  =  (/)  <->  ( P `  w )  =  (/) ) )
1312rexbidv 2478 . . . 4  |-  ( f  =  P  ->  ( E. w  e.  M  ( f `  w
)  =  (/)  <->  E. w  e.  M  ( P `  w )  =  (/) ) )
1411, 13imbi12d 234 . . 3  |-  ( f  =  P  ->  (
( -.  A. w  e.  M  ( f `  w )  =  1o 
->  E. w  e.  M  ( f `  w
)  =  (/) )  <->  ( -.  A. w  e.  M  ( P `  w )  =  1o  ->  E. w  e.  M  ( P `  w )  =  (/) ) ) )
15 fodjumkv.o . . . 4  |-  ( ph  ->  M  e. Markov )
16 ismkvmap 7145 . . . . 5  |-  ( M  e. Markov  ->  ( M  e. Markov  <->  A. f  e.  ( 2o 
^m  M ) ( -.  A. w  e.  M  ( f `  w )  =  1o 
->  E. w  e.  M  ( f `  w
)  =  (/) ) ) )
1716ibi 176 . . . 4  |-  ( M  e. Markov  ->  A. f  e.  ( 2o  ^m  M ) ( -.  A. w  e.  M  ( f `  w )  =  1o 
->  E. w  e.  M  ( f `  w
)  =  (/) ) )
1815, 17syl 14 . . 3  |-  ( ph  ->  A. f  e.  ( 2o  ^m  M ) ( -.  A. w  e.  M  ( f `  w )  =  1o 
->  E. w  e.  M  ( f `  w
)  =  (/) ) )
191, 3, 15fodjuf 7136 . . 3  |-  ( ph  ->  P  e.  ( 2o 
^m  M ) )
2014, 18, 19rspcdva 2846 . 2  |-  ( ph  ->  ( -.  A. w  e.  M  ( P `  w )  =  1o 
->  E. w  e.  M  ( P `  w )  =  (/) ) )
211adantr 276 . . . 4  |-  ( (
ph  /\  E. w  e.  M  ( P `  w )  =  (/) )  ->  F : M -onto->
( A B )
)
22 simpr 110 . . . . 5  |-  ( (
ph  /\  E. w  e.  M  ( P `  w )  =  (/) )  ->  E. w  e.  M  ( P `  w )  =  (/) )
23 fveqeq2 5519 . . . . . 6  |-  ( w  =  v  ->  (
( P `  w
)  =  (/)  <->  ( P `  v )  =  (/) ) )
2423cbvrexv 2704 . . . . 5  |-  ( E. w  e.  M  ( P `  w )  =  (/)  <->  E. v  e.  M  ( P `  v )  =  (/) )
2522, 24sylib 122 . . . 4  |-  ( (
ph  /\  E. w  e.  M  ( P `  w )  =  (/) )  ->  E. v  e.  M  ( P `  v )  =  (/) )
2621, 3, 25fodjum 7137 . . 3  |-  ( (
ph  /\  E. w  e.  M  ( P `  w )  =  (/) )  ->  E. x  x  e.  A )
2726ex 115 . 2  |-  ( ph  ->  ( E. w  e.  M  ( P `  w )  =  (/)  ->  E. x  x  e.  A ) )
287, 20, 273syld 57 1  |-  ( ph  ->  ( A  =/=  (/)  ->  E. x  x  e.  A )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 104    = wceq 1353   E.wex 1492    e. wcel 2148    =/= wne 2347   A.wral 2455   E.wrex 2456   (/)c0 3422   ifcif 3534    |-> cmpt 4061   -onto->wfo 5209   ` cfv 5211  (class class class)co 5868   1oc1o 6403   2oc2o 6404    ^m cmap 6641   ⊔ cdju 7029  inlcinl 7037  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-dc 835  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-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  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-mpt 4063  df-tr 4099  df-id 4289  df-iord 4362  df-on 4364  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-res 4634  df-ima 4635  df-iota 5173  df-fun 5213  df-fn 5214  df-f 5215  df-f1 5216  df-fo 5217  df-f1o 5218  df-fv 5219  df-ov 5871  df-oprab 5872  df-mpo 5873  df-1st 6134  df-2nd 6135  df-1o 6410  df-2o 6411  df-map 6643  df-dju 7030  df-inl 7039  df-inr 7040  df-markov 7143
This theorem is referenced by:  fodjumkv  7151
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