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Theorem fodjuomni 6783
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 5289 . . . . . . 7  |-  ( b  =  z  ->  (inl `  b )  =  (inl
`  z ) )
43eqeq2d 2099 . . . . . 6  |-  ( b  =  z  ->  (
( F `  a
)  =  (inl `  b )  <->  ( F `  a )  =  (inl
`  z ) ) )
54cbvrexv 2591 . . . . 5  |-  ( E. b  e.  A  ( F `  a )  =  (inl `  b
)  <->  E. z  e.  A  ( F `  a )  =  (inl `  z
) )
6 ifbi 3407 . . . . 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 7 . . . 4  |-  if ( E. b  e.  A  ( F `  a )  =  (inl `  b
) ,  (/) ,  1o )  =  if ( E. z  e.  A  ( F `  a )  =  (inl `  z
) ,  (/) ,  1o )
87mpteq2i 3917 . . 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 5289 . . . . . . 7  |-  ( a  =  y  ->  ( F `  a )  =  ( F `  y ) )
109eqeq1d 2096 . . . . . 6  |-  ( a  =  y  ->  (
( F `  a
)  =  (inl `  z )  <->  ( F `  y )  =  (inl
`  z ) ) )
1110rexbidv 2381 . . . . 5  |-  ( a  =  y  ->  ( E. z  e.  A  ( F `  a )  =  (inl `  z
)  <->  E. z  e.  A  ( F `  y )  =  (inl `  z
) ) )
1211ifbid 3408 . . . 4  |-  ( a  =  y  ->  if ( E. z  e.  A  ( F `  a )  =  (inl `  z
) ,  (/) ,  1o )  =  if ( E. z  e.  A  ( F `  y )  =  (inl `  z
) ,  (/) ,  1o ) )
1312cbvmptv 3926 . . 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 2108 . 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 6782 1  |-  ( ph  ->  ( E. x  x  e.  A  \/  A  =  (/) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103    \/ wo 664    = wceq 1289   E.wex 1426    e. wcel 1438   E.wrex 2360   (/)c0 3284   ifcif 3389    |-> cmpt 3891   -onto->wfo 5000   ` cfv 5002   1oc1o 6156   ⊔ cdju 6709  inlcinl 6716  Omnicomni 6767
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-nul 3957  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-rab 2368  df-v 2621  df-sbc 2839  df-csb 2932  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-if 3390  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-br 3838  df-opab 3892  df-mpt 3893  df-tr 3929  df-id 4111  df-iord 4184  df-on 4186  df-suc 4189  df-iom 4396  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-fo 5008  df-fv 5010  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-1st 5893  df-2nd 5894  df-1o 6163  df-2o 6164  df-map 6387  df-dju 6710  df-inl 6718  df-inr 6719  df-omni 6769
This theorem is referenced by:  exmidsbthrlem  11569
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