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Theorem fodjuomni 7125
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 5496 . . . . . . 7  |-  ( b  =  z  ->  (inl `  b )  =  (inl
`  z ) )
43eqeq2d 2182 . . . . . 6  |-  ( b  =  z  ->  (
( F `  a
)  =  (inl `  b )  <->  ( F `  a )  =  (inl
`  z ) ) )
54cbvrexv 2697 . . . . 5  |-  ( E. b  e.  A  ( F `  a )  =  (inl `  b
)  <->  E. z  e.  A  ( F `  a )  =  (inl `  z
) )
6 ifbi 3546 . . . . 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 4076 . . 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 5496 . . . . . . 7  |-  ( a  =  y  ->  ( F `  a )  =  ( F `  y ) )
109eqeq1d 2179 . . . . . 6  |-  ( a  =  y  ->  (
( F `  a
)  =  (inl `  z )  <->  ( F `  y )  =  (inl
`  z ) ) )
1110rexbidv 2471 . . . . 5  |-  ( a  =  y  ->  ( E. z  e.  A  ( F `  a )  =  (inl `  z
)  <->  E. z  e.  A  ( F `  y )  =  (inl `  z
) ) )
1211ifbid 3547 . . . 4  |-  ( a  =  y  ->  if ( E. z  e.  A  ( F `  a )  =  (inl `  z
) ,  (/) ,  1o )  =  if ( E. z  e.  A  ( F `  y )  =  (inl `  z
) ,  (/) ,  1o ) )
1312cbvmptv 4085 . . 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 2191 . 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 7124 1  |-  ( ph  ->  ( E. x  x  e.  A  \/  A  =  (/) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    \/ wo 703    = wceq 1348   E.wex 1485    e. wcel 2141   E.wrex 2449   (/)c0 3414   ifcif 3526    |-> cmpt 4050   -onto->wfo 5196   ` cfv 5198   1oc1o 6388   ⊔ cdju 7014  inlcinl 7022  Omnicomni 7110
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521
This theorem depends on definitions:  df-bi 116  df-dc 830  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-iord 4351  df-on 4353  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-1o 6395  df-2o 6396  df-map 6628  df-dju 7015  df-inl 7024  df-inr 7025  df-omni 7111
This theorem is referenced by:  ctssexmid  7126  exmidunben  12381  exmidsbthrlem  14054  sbthomlem  14057
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