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Theorem fodju0 7023
Description: Lemma for fodjuomni 7025 and fodjumkv 7038. A condition which shows that  A is empty. (Contributed by Jim Kingdon, 27-Jul-2022.) (Revised by Jim Kingdon, 25-Mar-2023.)
Hypotheses
Ref Expression
fodjuf.fo  |-  ( ph  ->  F : O -onto-> ( A B ) )
fodjuf.p  |-  P  =  ( y  e.  O  |->  if ( E. z  e.  A  ( F `  y )  =  (inl
`  z ) ,  (/) ,  1o ) )
fodju0.1  |-  ( ph  ->  A. w  e.  O  ( P `  w )  =  1o )
Assertion
Ref Expression
fodju0  |-  ( ph  ->  A  =  (/) )
Distinct variable groups:    ph, y, z   
y, O, z    z, A    z, B    z, F    y, A    y, F    w, O    w, P
Allowed substitution hints:    ph( w)    A( w)    B( y, w)    P( y,
z)    F( w)

Proof of Theorem fodju0
Dummy variables  u  v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fodjuf.fo . . . . 5  |-  ( ph  ->  F : O -onto-> ( A B ) )
2 djulcl 6940 . . . . 5  |-  ( u  e.  A  ->  (inl `  u )  e.  ( A B ) )
3 foelrn 5658 . . . . 5  |-  ( ( F : O -onto-> ( A B )  /\  (inl `  u )  e.  ( A B ) )  ->  E. v  e.  O  (inl `  u )  =  ( F `  v
) )
41, 2, 3syl2an 287 . . . 4  |-  ( (
ph  /\  u  e.  A )  ->  E. v  e.  O  (inl `  u
)  =  ( F `
 v ) )
5 fodjuf.p . . . . . 6  |-  P  =  ( y  e.  O  |->  if ( E. z  e.  A  ( F `  y )  =  (inl
`  z ) ,  (/) ,  1o ) )
6 fveqeq2 5434 . . . . . . . 8  |-  ( y  =  v  ->  (
( F `  y
)  =  (inl `  z )  <->  ( F `  v )  =  (inl
`  z ) ) )
76rexbidv 2439 . . . . . . 7  |-  ( y  =  v  ->  ( E. z  e.  A  ( F `  y )  =  (inl `  z
)  <->  E. z  e.  A  ( F `  v )  =  (inl `  z
) ) )
87ifbid 3494 . . . . . 6  |-  ( y  =  v  ->  if ( E. z  e.  A  ( F `  y )  =  (inl `  z
) ,  (/) ,  1o )  =  if ( E. z  e.  A  ( F `  v )  =  (inl `  z
) ,  (/) ,  1o ) )
9 simprl 521 . . . . . 6  |-  ( ( ( ph  /\  u  e.  A )  /\  (
v  e.  O  /\  (inl `  u )  =  ( F `  v
) ) )  -> 
v  e.  O )
10 peano1 4512 . . . . . . . 8  |-  (/)  e.  om
1110a1i 9 . . . . . . 7  |-  ( ( ( ph  /\  u  e.  A )  /\  (
v  e.  O  /\  (inl `  u )  =  ( F `  v
) ) )  ->  (/) 
e.  om )
12 1onn 6420 . . . . . . . 8  |-  1o  e.  om
1312a1i 9 . . . . . . 7  |-  ( ( ( ph  /\  u  e.  A )  /\  (
v  e.  O  /\  (inl `  u )  =  ( F `  v
) ) )  ->  1o  e.  om )
141fodjuomnilemdc 7020 . . . . . . . 8  |-  ( (
ph  /\  v  e.  O )  -> DECID  E. z  e.  A  ( F `  v )  =  (inl `  z
) )
1514ad2ant2r 501 . . . . . . 7  |-  ( ( ( ph  /\  u  e.  A )  /\  (
v  e.  O  /\  (inl `  u )  =  ( F `  v
) ) )  -> DECID  E. z  e.  A  ( F `  v )  =  (inl
`  z ) )
1611, 13, 15ifcldcd 3508 . . . . . 6  |-  ( ( ( ph  /\  u  e.  A )  /\  (
v  e.  O  /\  (inl `  u )  =  ( F `  v
) ) )  ->  if ( E. z  e.  A  ( F `  v )  =  (inl
`  z ) ,  (/) ,  1o )  e. 
om )
175, 8, 9, 16fvmptd3 5518 . . . . 5  |-  ( ( ( ph  /\  u  e.  A )  /\  (
v  e.  O  /\  (inl `  u )  =  ( F `  v
) ) )  -> 
( P `  v
)  =  if ( E. z  e.  A  ( F `  v )  =  (inl `  z
) ,  (/) ,  1o ) )
18 fveqeq2 5434 . . . . . 6  |-  ( w  =  v  ->  (
( P `  w
)  =  1o  <->  ( P `  v )  =  1o ) )
19 fodju0.1 . . . . . . 7  |-  ( ph  ->  A. w  e.  O  ( P `  w )  =  1o )
2019ad2antrr 480 . . . . . 6  |-  ( ( ( ph  /\  u  e.  A )  /\  (
v  e.  O  /\  (inl `  u )  =  ( F `  v
) ) )  ->  A. w  e.  O  ( P `  w )  =  1o )
2118, 20, 9rspcdva 2795 . . . . 5  |-  ( ( ( ph  /\  u  e.  A )  /\  (
v  e.  O  /\  (inl `  u )  =  ( F `  v
) ) )  -> 
( P `  v
)  =  1o )
22 simplr 520 . . . . . . 7  |-  ( ( ( ph  /\  u  e.  A )  /\  (
v  e.  O  /\  (inl `  u )  =  ( F `  v
) ) )  ->  u  e.  A )
23 simprr 522 . . . . . . . 8  |-  ( ( ( ph  /\  u  e.  A )  /\  (
v  e.  O  /\  (inl `  u )  =  ( F `  v
) ) )  -> 
(inl `  u )  =  ( F `  v ) )
2423eqcomd 2146 . . . . . . 7  |-  ( ( ( ph  /\  u  e.  A )  /\  (
v  e.  O  /\  (inl `  u )  =  ( F `  v
) ) )  -> 
( F `  v
)  =  (inl `  u ) )
25 fveq2 5425 . . . . . . . 8  |-  ( z  =  u  ->  (inl `  z )  =  (inl
`  u ) )
2625rspceeqv 2808 . . . . . . 7  |-  ( ( u  e.  A  /\  ( F `  v )  =  (inl `  u
) )  ->  E. z  e.  A  ( F `  v )  =  (inl
`  z ) )
2722, 24, 26syl2anc 409 . . . . . 6  |-  ( ( ( ph  /\  u  e.  A )  /\  (
v  e.  O  /\  (inl `  u )  =  ( F `  v
) ) )  ->  E. z  e.  A  ( F `  v )  =  (inl `  z
) )
2827iftrued 3482 . . . . 5  |-  ( ( ( ph  /\  u  e.  A )  /\  (
v  e.  O  /\  (inl `  u )  =  ( F `  v
) ) )  ->  if ( E. z  e.  A  ( F `  v )  =  (inl
`  z ) ,  (/) ,  1o )  =  (/) )
2917, 21, 283eqtr3rd 2182 . . . 4  |-  ( ( ( ph  /\  u  e.  A )  /\  (
v  e.  O  /\  (inl `  u )  =  ( F `  v
) ) )  ->  (/)  =  1o )
304, 29rexlimddv 2555 . . 3  |-  ( (
ph  /\  u  e.  A )  ->  (/)  =  1o )
31 1n0 6333 . . . . 5  |-  1o  =/=  (/)
3231nesymi 2355 . . . 4  |-  -.  (/)  =  1o
3332a1i 9 . . 3  |-  ( (
ph  /\  u  e.  A )  ->  -.  (/)  =  1o )
3430, 33pm2.65da 651 . 2  |-  ( ph  ->  -.  u  e.  A
)
3534eq0rdv 3408 1  |-  ( ph  ->  A  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103  DECID wdc 820    = wceq 1332    e. wcel 1481   A.wral 2417   E.wrex 2418   (/)c0 3364   ifcif 3475    |-> cmpt 3993   omcom 4508   -onto->wfo 5125   ` cfv 5127   1oc1o 6310   ⊔ cdju 6926  inlcinl 6934
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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4050  ax-nul 4058  ax-pow 4102  ax-pr 4135  ax-un 4359
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-v 2689  df-sbc 2911  df-csb 3005  df-dif 3074  df-un 3076  df-in 3078  df-ss 3085  df-nul 3365  df-if 3476  df-pw 3513  df-sn 3534  df-pr 3535  df-op 3537  df-uni 3741  df-int 3776  df-br 3934  df-opab 3994  df-mpt 3995  df-tr 4031  df-id 4219  df-iord 4292  df-on 4294  df-suc 4297  df-iom 4509  df-xp 4549  df-rel 4550  df-cnv 4551  df-co 4552  df-dm 4553  df-rn 4554  df-res 4555  df-ima 4556  df-iota 5092  df-fun 5129  df-fn 5130  df-f 5131  df-f1 5132  df-fo 5133  df-f1o 5134  df-fv 5135  df-1st 6042  df-2nd 6043  df-1o 6317  df-dju 6927  df-inl 6936  df-inr 6937
This theorem is referenced by:  fodjuomnilemres  7024  fodjumkvlemres  7037
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