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Theorem fodjuomnilem0 6781
Description: Lemma for fodjuomni 6783. The case where A is empty. (Contributed by Jim Kingdon, 27-Jul-2022.)
Hypotheses
Ref Expression
fodjuomni.o  |-  ( ph  ->  O  e. Omni )
fodjuomni.fo  |-  ( ph  ->  F : O -onto-> ( A B ) )
fodjuomni.p  |-  P  =  ( y  e.  O  |->  if ( E. z  e.  A  ( F `  y )  =  (inl
`  z ) ,  (/) ,  1o ) )
fodjuomni.1  |-  ( ph  ->  A. w  e.  O  ( P `  w )  =  1o )
Assertion
Ref Expression
fodjuomnilem0  |-  ( 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 fodjuomnilem0
Dummy variables  u  v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fodjuomni.fo . . . . 5  |-  ( ph  ->  F : O -onto-> ( A B ) )
2 djulcl 6722 . . . . 5  |-  ( u  e.  A  ->  (inl `  u )  e.  ( A B ) )
3 foelrn 5513 . . . . 5  |-  ( ( F : O -onto-> ( A B )  /\  (inl `  u )  e.  ( A B ) )  ->  E. v  e.  O  (inl `  u )  =  ( F `  v
) )
41, 2, 3syl2an 283 . . . 4  |-  ( (
ph  /\  u  e.  A )  ->  E. v  e.  O  (inl `  u
)  =  ( F `
 v ) )
5 simprl 498 . . . . . 6  |-  ( ( ( ph  /\  u  e.  A )  /\  (
v  e.  O  /\  (inl `  u )  =  ( F `  v
) ) )  -> 
v  e.  O )
6 peano1 4399 . . . . . . . 8  |-  (/)  e.  om
76a1i 9 . . . . . . 7  |-  ( ( ( ph  /\  u  e.  A )  /\  (
v  e.  O  /\  (inl `  u )  =  ( F `  v
) ) )  ->  (/) 
e.  om )
8 1onn 6259 . . . . . . . 8  |-  1o  e.  om
98a1i 9 . . . . . . 7  |-  ( ( ( ph  /\  u  e.  A )  /\  (
v  e.  O  /\  (inl `  u )  =  ( F `  v
) ) )  ->  1o  e.  om )
101fodjuomnilemdc 6778 . . . . . . . 8  |-  ( (
ph  /\  v  e.  O )  -> DECID  E. z  e.  A  ( F `  v )  =  (inl `  z
) )
1110ad2ant2r 493 . . . . . . 7  |-  ( ( ( ph  /\  u  e.  A )  /\  (
v  e.  O  /\  (inl `  u )  =  ( F `  v
) ) )  -> DECID  E. z  e.  A  ( F `  v )  =  (inl
`  z ) )
127, 9, 11ifcldcd 3422 . . . . . 6  |-  ( ( ( ph  /\  u  e.  A )  /\  (
v  e.  O  /\  (inl `  u )  =  ( F `  v
) ) )  ->  if ( E. z  e.  A  ( F `  v )  =  (inl
`  z ) ,  (/) ,  1o )  e. 
om )
13 fveq2 5289 . . . . . . . . . 10  |-  ( y  =  v  ->  ( F `  y )  =  ( F `  v ) )
1413eqeq1d 2096 . . . . . . . . 9  |-  ( y  =  v  ->  (
( F `  y
)  =  (inl `  z )  <->  ( F `  v )  =  (inl
`  z ) ) )
1514rexbidv 2381 . . . . . . . 8  |-  ( y  =  v  ->  ( E. z  e.  A  ( F `  y )  =  (inl `  z
)  <->  E. z  e.  A  ( F `  v )  =  (inl `  z
) ) )
1615ifbid 3408 . . . . . . 7  |-  ( y  =  v  ->  if ( E. z  e.  A  ( F `  y )  =  (inl `  z
) ,  (/) ,  1o )  =  if ( E. z  e.  A  ( F `  v )  =  (inl `  z
) ,  (/) ,  1o ) )
17 fodjuomni.p . . . . . . 7  |-  P  =  ( y  e.  O  |->  if ( E. z  e.  A  ( F `  y )  =  (inl
`  z ) ,  (/) ,  1o ) )
1816, 17fvmptg 5364 . . . . . 6  |-  ( ( v  e.  O  /\  if ( E. z  e.  A  ( F `  v )  =  (inl
`  z ) ,  (/) ,  1o )  e. 
om )  ->  ( P `  v )  =  if ( E. z  e.  A  ( F `  v )  =  (inl
`  z ) ,  (/) ,  1o ) )
195, 12, 18syl2anc 403 . . . . 5  |-  ( ( ( ph  /\  u  e.  A )  /\  (
v  e.  O  /\  (inl `  u )  =  ( F `  v
) ) )  -> 
( P `  v
)  =  if ( E. z  e.  A  ( F `  v )  =  (inl `  z
) ,  (/) ,  1o ) )
20 fveq2 5289 . . . . . . 7  |-  ( w  =  v  ->  ( P `  w )  =  ( P `  v ) )
2120eqeq1d 2096 . . . . . 6  |-  ( w  =  v  ->  (
( P `  w
)  =  1o  <->  ( P `  v )  =  1o ) )
22 fodjuomni.1 . . . . . . 7  |-  ( ph  ->  A. w  e.  O  ( P `  w )  =  1o )
2322ad2antrr 472 . . . . . 6  |-  ( ( ( ph  /\  u  e.  A )  /\  (
v  e.  O  /\  (inl `  u )  =  ( F `  v
) ) )  ->  A. w  e.  O  ( P `  w )  =  1o )
2421, 23, 5rspcdva 2727 . . . . 5  |-  ( ( ( ph  /\  u  e.  A )  /\  (
v  e.  O  /\  (inl `  u )  =  ( F `  v
) ) )  -> 
( P `  v
)  =  1o )
25 simplr 497 . . . . . . 7  |-  ( ( ( ph  /\  u  e.  A )  /\  (
v  e.  O  /\  (inl `  u )  =  ( F `  v
) ) )  ->  u  e.  A )
26 simprr 499 . . . . . . . 8  |-  ( ( ( ph  /\  u  e.  A )  /\  (
v  e.  O  /\  (inl `  u )  =  ( F `  v
) ) )  -> 
(inl `  u )  =  ( F `  v ) )
2726eqcomd 2093 . . . . . . 7  |-  ( ( ( ph  /\  u  e.  A )  /\  (
v  e.  O  /\  (inl `  u )  =  ( F `  v
) ) )  -> 
( F `  v
)  =  (inl `  u ) )
28 fveq2 5289 . . . . . . . . 9  |-  ( z  =  u  ->  (inl `  z )  =  (inl
`  u ) )
2928eqeq2d 2099 . . . . . . . 8  |-  ( z  =  u  ->  (
( F `  v
)  =  (inl `  z )  <->  ( F `  v )  =  (inl
`  u ) ) )
3029rspcev 2722 . . . . . . 7  |-  ( ( u  e.  A  /\  ( F `  v )  =  (inl `  u
) )  ->  E. z  e.  A  ( F `  v )  =  (inl
`  z ) )
3125, 27, 30syl2anc 403 . . . . . 6  |-  ( ( ( ph  /\  u  e.  A )  /\  (
v  e.  O  /\  (inl `  u )  =  ( F `  v
) ) )  ->  E. z  e.  A  ( F `  v )  =  (inl `  z
) )
3231iftrued 3396 . . . . 5  |-  ( ( ( ph  /\  u  e.  A )  /\  (
v  e.  O  /\  (inl `  u )  =  ( F `  v
) ) )  ->  if ( E. z  e.  A  ( F `  v )  =  (inl
`  z ) ,  (/) ,  1o )  =  (/) )
3319, 24, 323eqtr3rd 2129 . . . 4  |-  ( ( ( ph  /\  u  e.  A )  /\  (
v  e.  O  /\  (inl `  u )  =  ( F `  v
) ) )  ->  (/)  =  1o )
344, 33rexlimddv 2493 . . 3  |-  ( (
ph  /\  u  e.  A )  ->  (/)  =  1o )
35 1n0 6179 . . . . 5  |-  1o  =/=  (/)
3635nesymi 2301 . . . 4  |-  -.  (/)  =  1o
3736a1i 9 . . 3  |-  ( (
ph  /\  u  e.  A )  ->  -.  (/)  =  1o )
3834, 37pm2.65da 622 . 2  |-  ( ph  ->  -.  u  e.  A
)
3938eq0rdv 3324 1  |-  ( ph  ->  A  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102  DECID wdc 780    = wceq 1289    e. wcel 1438   A.wral 2359   E.wrex 2360   (/)c0 3284   ifcif 3389    |-> cmpt 3891   omcom 4395   -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
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-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-1st 5893  df-2nd 5894  df-1o 6163  df-dju 6710  df-inl 6718  df-inr 6719
This theorem is referenced by:  fodjuomnilemres  6782
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