ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  fodjuomni Unicode version

Theorem fodjuomni 7147
Description: A condition which ensures  A is either inhabited or empty. Lemma 3.2 of [PradicBrown2022], p. 4. (Contributed by Jim Kingdon, 27-Jul-2022.)
Hypotheses
Ref Expression
fodjuomni.o  |-  ( ph  ->  O  e. Omni )
fodjuomni.fo  |-  ( ph  ->  F : O -onto-> ( A B ) )
Assertion
Ref Expression
fodjuomni  |-  ( ph  ->  ( E. x  x  e.  A  \/  A  =  (/) ) )
Distinct variable group:    x, A
Allowed substitution hints:    ph( x)    B( x)    F( x)    O( x)

Proof of Theorem fodjuomni
Dummy variables  a  b  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fodjuomni.o . 2  |-  ( ph  ->  O  e. Omni )
2 fodjuomni.fo . 2  |-  ( ph  ->  F : O -onto-> ( A B ) )
3 fveq2 5516 . . . . . . 7  |-  ( b  =  z  ->  (inl `  b )  =  (inl
`  z ) )
43eqeq2d 2189 . . . . . 6  |-  ( b  =  z  ->  (
( F `  a
)  =  (inl `  b )  <->  ( F `  a )  =  (inl
`  z ) ) )
54cbvrexv 2705 . . . . 5  |-  ( E. b  e.  A  ( F `  a )  =  (inl `  b
)  <->  E. z  e.  A  ( F `  a )  =  (inl `  z
) )
6 ifbi 3555 . . . . 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 4091 . . 3  |-  ( a  e.  O  |->  if ( E. b  e.  A  ( F `  a )  =  (inl `  b
) ,  (/) ,  1o ) )  =  ( a  e.  O  |->  if ( E. z  e.  A  ( F `  a )  =  (inl
`  z ) ,  (/) ,  1o ) )
9 fveq2 5516 . . . . . . 7  |-  ( a  =  y  ->  ( F `  a )  =  ( F `  y ) )
109eqeq1d 2186 . . . . . 6  |-  ( a  =  y  ->  (
( F `  a
)  =  (inl `  z )  <->  ( F `  y )  =  (inl
`  z ) ) )
1110rexbidv 2478 . . . . 5  |-  ( a  =  y  ->  ( E. z  e.  A  ( F `  a )  =  (inl `  z
)  <->  E. z  e.  A  ( F `  y )  =  (inl `  z
) ) )
1211ifbid 3556 . . . 4  |-  ( a  =  y  ->  if ( E. z  e.  A  ( F `  a )  =  (inl `  z
) ,  (/) ,  1o )  =  if ( E. z  e.  A  ( F `  y )  =  (inl `  z
) ,  (/) ,  1o ) )
1312cbvmptv 4100 . . 3  |-  ( a  e.  O  |->  if ( E. z  e.  A  ( F `  a )  =  (inl `  z
) ,  (/) ,  1o ) )  =  ( y  e.  O  |->  if ( E. z  e.  A  ( F `  y )  =  (inl
`  z ) ,  (/) ,  1o ) )
148, 13eqtri 2198 . 2  |-  ( a  e.  O  |->  if ( E. b  e.  A  ( F `  a )  =  (inl `  b
) ,  (/) ,  1o ) )  =  ( y  e.  O  |->  if ( E. z  e.  A  ( F `  y )  =  (inl
`  z ) ,  (/) ,  1o ) )
151, 2, 14fodjuomnilemres 7146 1  |-  ( ph  ->  ( E. x  x  e.  A  \/  A  =  (/) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 105    \/ wo 708    = wceq 1353   E.wex 1492    e. wcel 2148   E.wrex 2456   (/)c0 3423   ifcif 3535    |-> cmpt 4065   -onto->wfo 5215   ` cfv 5217   1oc1o 6410   ⊔ cdju 7036  inlcinl 7044  Omnicomni 7132
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 4122  ax-nul 4130  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537
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 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-if 3536  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-br 4005  df-opab 4066  df-mpt 4067  df-tr 4103  df-id 4294  df-iord 4367  df-on 4369  df-suc 4372  df-iom 4591  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-ov 5878  df-oprab 5879  df-mpo 5880  df-1st 6141  df-2nd 6142  df-1o 6417  df-2o 6418  df-map 6650  df-dju 7037  df-inl 7046  df-inr 7047  df-omni 7133
This theorem is referenced by:  ctssexmid  7148  exmidunben  12427  exmidsbthrlem  14773  sbthomlem  14776
  Copyright terms: Public domain W3C validator