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Theorem fodjuomni 7177
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 5534 . . . . . . 7  |-  ( b  =  z  ->  (inl `  b )  =  (inl
`  z ) )
43eqeq2d 2201 . . . . . 6  |-  ( b  =  z  ->  (
( F `  a
)  =  (inl `  b )  <->  ( F `  a )  =  (inl
`  z ) ) )
54cbvrexv 2719 . . . . 5  |-  ( E. b  e.  A  ( F `  a )  =  (inl `  b
)  <->  E. z  e.  A  ( F `  a )  =  (inl `  z
) )
6 ifbi 3569 . . . . 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 4105 . . 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 5534 . . . . . . 7  |-  ( a  =  y  ->  ( F `  a )  =  ( F `  y ) )
109eqeq1d 2198 . . . . . 6  |-  ( a  =  y  ->  (
( F `  a
)  =  (inl `  z )  <->  ( F `  y )  =  (inl
`  z ) ) )
1110rexbidv 2491 . . . . 5  |-  ( a  =  y  ->  ( E. z  e.  A  ( F `  a )  =  (inl `  z
)  <->  E. z  e.  A  ( F `  y )  =  (inl `  z
) ) )
1211ifbid 3570 . . . 4  |-  ( a  =  y  ->  if ( E. z  e.  A  ( F `  a )  =  (inl `  z
) ,  (/) ,  1o )  =  if ( E. z  e.  A  ( F `  y )  =  (inl `  z
) ,  (/) ,  1o ) )
1312cbvmptv 4114 . . 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 2210 . 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 7176 1  |-  ( ph  ->  ( E. x  x  e.  A  \/  A  =  (/) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 105    \/ wo 709    = wceq 1364   E.wex 1503    e. wcel 2160   E.wrex 2469   (/)c0 3437   ifcif 3549    |-> cmpt 4079   -onto->wfo 5233   ` cfv 5235   1oc1o 6434   ⊔ cdju 7066  inlcinl 7074  Omnicomni 7162
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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-if 3550  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4311  df-iord 4384  df-on 4386  df-suc 4389  df-iom 4608  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-ov 5899  df-oprab 5900  df-mpo 5901  df-1st 6165  df-2nd 6166  df-1o 6441  df-2o 6442  df-map 6676  df-dju 7067  df-inl 7076  df-inr 7077  df-omni 7163
This theorem is referenced by:  ctssexmid  7178  exmidunben  12477  exmidsbthrlem  15229  sbthomlem  15232
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