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Theorem fodjumkv 7151
Description: A condition which ensures that a nonempty set is inhabited. (Contributed by Jim Kingdon, 25-Mar-2023.)
Hypotheses
Ref Expression
fodjumkv.o  |-  ( ph  ->  M  e. Markov )
fodjumkv.fo  |-  ( ph  ->  F : M -onto-> ( A B ) )
Assertion
Ref Expression
fodjumkv  |-  ( ph  ->  ( A  =/=  (/)  ->  E. x  x  e.  A )
)
Distinct variable group:    x, A
Allowed substitution hints:    ph( x)    B( x)    F( x)    M( x)

Proof of Theorem fodjumkv
Dummy variables  a  b  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fodjumkv.o . 2  |-  ( ph  ->  M  e. Markov )
2 fodjumkv.fo . 2  |-  ( ph  ->  F : M -onto-> ( A B ) )
3 fveq2 5510 . . . . . . 7  |-  ( b  =  z  ->  (inl `  b )  =  (inl
`  z ) )
43eqeq2d 2189 . . . . . 6  |-  ( b  =  z  ->  (
( F `  a
)  =  (inl `  b )  <->  ( F `  a )  =  (inl
`  z ) ) )
54cbvrexv 2704 . . . . 5  |-  ( E. b  e.  A  ( F `  a )  =  (inl `  b
)  <->  E. z  e.  A  ( F `  a )  =  (inl `  z
) )
6 ifbi 3554 . . . . 5  |-  ( ( E. b  e.  A  ( F `  a )  =  (inl `  b
)  <->  E. z  e.  A  ( F `  a )  =  (inl `  z
) )  ->  if ( E. b  e.  A  ( F `  a )  =  (inl `  b
) ,  (/) ,  1o )  =  if ( E. z  e.  A  ( F `  a )  =  (inl `  z
) ,  (/) ,  1o ) )
75, 6ax-mp 5 . . . 4  |-  if ( E. b  e.  A  ( F `  a )  =  (inl `  b
) ,  (/) ,  1o )  =  if ( E. z  e.  A  ( F `  a )  =  (inl `  z
) ,  (/) ,  1o )
87mpteq2i 4087 . . 3  |-  ( a  e.  M  |->  if ( E. b  e.  A  ( F `  a )  =  (inl `  b
) ,  (/) ,  1o ) )  =  ( a  e.  M  |->  if ( E. z  e.  A  ( F `  a )  =  (inl
`  z ) ,  (/) ,  1o ) )
9 fveqeq2 5519 . . . . . 6  |-  ( a  =  y  ->  (
( F `  a
)  =  (inl `  z )  <->  ( F `  y )  =  (inl
`  z ) ) )
109rexbidv 2478 . . . . 5  |-  ( a  =  y  ->  ( E. z  e.  A  ( F `  a )  =  (inl `  z
)  <->  E. z  e.  A  ( F `  y )  =  (inl `  z
) ) )
1110ifbid 3555 . . . 4  |-  ( a  =  y  ->  if ( E. z  e.  A  ( F `  a )  =  (inl `  z
) ,  (/) ,  1o )  =  if ( E. z  e.  A  ( F `  y )  =  (inl `  z
) ,  (/) ,  1o ) )
1211cbvmptv 4096 . . 3  |-  ( a  e.  M  |->  if ( E. z  e.  A  ( F `  a )  =  (inl `  z
) ,  (/) ,  1o ) )  =  ( y  e.  M  |->  if ( E. z  e.  A  ( F `  y )  =  (inl
`  z ) ,  (/) ,  1o ) )
138, 12eqtri 2198 . 2  |-  ( a  e.  M  |->  if ( E. b  e.  A  ( F `  a )  =  (inl `  b
) ,  (/) ,  1o ) )  =  ( y  e.  M  |->  if ( E. z  e.  A  ( F `  y )  =  (inl
`  z ) ,  (/) ,  1o ) )
141, 2, 13fodjumkvlemres 7150 1  |-  ( ph  ->  ( A  =/=  (/)  ->  E. x  x  e.  A )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 105    = wceq 1353   E.wex 1492    e. wcel 2148    =/= wne 2347   E.wrex 2456   (/)c0 3422   ifcif 3534    |-> cmpt 4061   -onto->wfo 5209   ` cfv 5211   1oc1o 6403   ⊔ 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: (None)
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