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Theorem fodjuomnilemm 6780
Description: Lemma for fodjuomni 6783. The case where A is inhabited. (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.z  |-  ( ph  ->  E. w  e.  O  ( P `  w )  =  (/) )
Assertion
Ref Expression
fodjuomnilemm  |-  ( ph  ->  E. x  x  e.  A )
Distinct variable groups:    ph, y, z   
y, O, z    z, A    z, B    z, F    w, A, x, z    y, A, w    y, F    ph, w
Allowed substitution hints:    ph( x)    B( x, y, w)    P( x, y, z, w)    F( x, w)    O( x, w)

Proof of Theorem fodjuomnilemm
StepHypRef Expression
1 fodjuomni.z . 2  |-  ( ph  ->  E. w  e.  O  ( P `  w )  =  (/) )
2 1n0 6179 . . . . . . . . 9  |-  1o  =/=  (/)
32nesymi 2301 . . . . . . . 8  |-  -.  (/)  =  1o
43intnan 876 . . . . . . 7  |-  -.  ( -.  E. z  e.  A  ( F `  w )  =  (inl `  z
)  /\  (/)  =  1o )
54a1i 9 . . . . . 6  |-  ( (
ph  /\  ( w  e.  O  /\  ( P `  w )  =  (/) ) )  ->  -.  ( -.  E. z  e.  A  ( F `  w )  =  (inl
`  z )  /\  (/)  =  1o ) )
6 simprr 499 . . . . . . . 8  |-  ( (
ph  /\  ( w  e.  O  /\  ( P `  w )  =  (/) ) )  -> 
( P `  w
)  =  (/) )
7 simprl 498 . . . . . . . . 9  |-  ( (
ph  /\  ( w  e.  O  /\  ( P `  w )  =  (/) ) )  ->  w  e.  O )
8 peano1 4399 . . . . . . . . . . 11  |-  (/)  e.  om
98a1i 9 . . . . . . . . . 10  |-  ( (
ph  /\  ( w  e.  O  /\  ( P `  w )  =  (/) ) )  ->  (/) 
e.  om )
10 1onn 6259 . . . . . . . . . . 11  |-  1o  e.  om
1110a1i 9 . . . . . . . . . 10  |-  ( (
ph  /\  ( w  e.  O  /\  ( P `  w )  =  (/) ) )  ->  1o  e.  om )
12 fodjuomni.fo . . . . . . . . . . . 12  |-  ( ph  ->  F : O -onto-> ( A B ) )
1312fodjuomnilemdc 6778 . . . . . . . . . . 11  |-  ( (
ph  /\  w  e.  O )  -> DECID  E. z  e.  A  ( F `  w )  =  (inl `  z
) )
1413adantrr 463 . . . . . . . . . 10  |-  ( (
ph  /\  ( w  e.  O  /\  ( P `  w )  =  (/) ) )  -> DECID  E. z  e.  A  ( F `  w )  =  (inl
`  z ) )
159, 11, 14ifcldcd 3422 . . . . . . . . 9  |-  ( (
ph  /\  ( w  e.  O  /\  ( P `  w )  =  (/) ) )  ->  if ( E. z  e.  A  ( F `  w )  =  (inl
`  z ) ,  (/) ,  1o )  e. 
om )
16 fveq2 5289 . . . . . . . . . . . . 13  |-  ( y  =  w  ->  ( F `  y )  =  ( F `  w ) )
1716eqeq1d 2096 . . . . . . . . . . . 12  |-  ( y  =  w  ->  (
( F `  y
)  =  (inl `  z )  <->  ( F `  w )  =  (inl
`  z ) ) )
1817rexbidv 2381 . . . . . . . . . . 11  |-  ( y  =  w  ->  ( E. z  e.  A  ( F `  y )  =  (inl `  z
)  <->  E. z  e.  A  ( F `  w )  =  (inl `  z
) ) )
1918ifbid 3408 . . . . . . . . . 10  |-  ( y  =  w  ->  if ( E. z  e.  A  ( F `  y )  =  (inl `  z
) ,  (/) ,  1o )  =  if ( E. z  e.  A  ( F `  w )  =  (inl `  z
) ,  (/) ,  1o ) )
20 fodjuomni.p . . . . . . . . . 10  |-  P  =  ( y  e.  O  |->  if ( E. z  e.  A  ( F `  y )  =  (inl
`  z ) ,  (/) ,  1o ) )
2119, 20fvmptg 5364 . . . . . . . . 9  |-  ( ( w  e.  O  /\  if ( E. z  e.  A  ( F `  w )  =  (inl
`  z ) ,  (/) ,  1o )  e. 
om )  ->  ( P `  w )  =  if ( E. z  e.  A  ( F `  w )  =  (inl
`  z ) ,  (/) ,  1o ) )
227, 15, 21syl2anc 403 . . . . . . . 8  |-  ( (
ph  /\  ( w  e.  O  /\  ( P `  w )  =  (/) ) )  -> 
( P `  w
)  =  if ( E. z  e.  A  ( F `  w )  =  (inl `  z
) ,  (/) ,  1o ) )
236, 22eqtr3d 2122 . . . . . . 7  |-  ( (
ph  /\  ( w  e.  O  /\  ( P `  w )  =  (/) ) )  ->  (/)  =  if ( E. z  e.  A  ( F `  w )  =  (inl `  z
) ,  (/) ,  1o ) )
24 eqifdc 3421 . . . . . . . 8  |-  (DECID  E. z  e.  A  ( F `  w )  =  (inl
`  z )  -> 
( (/)  =  if ( E. z  e.  A  ( F `  w )  =  (inl `  z
) ,  (/) ,  1o ) 
<->  ( ( E. z  e.  A  ( F `  w )  =  (inl
`  z )  /\  (/)  =  (/) )  \/  ( -.  E. z  e.  A  ( F `  w )  =  (inl `  z
)  /\  (/)  =  1o ) ) ) )
2514, 24syl 14 . . . . . . 7  |-  ( (
ph  /\  ( w  e.  O  /\  ( P `  w )  =  (/) ) )  -> 
( (/)  =  if ( E. z  e.  A  ( F `  w )  =  (inl `  z
) ,  (/) ,  1o ) 
<->  ( ( E. z  e.  A  ( F `  w )  =  (inl
`  z )  /\  (/)  =  (/) )  \/  ( -.  E. z  e.  A  ( F `  w )  =  (inl `  z
)  /\  (/)  =  1o ) ) ) )
2623, 25mpbid 145 . . . . . 6  |-  ( (
ph  /\  ( w  e.  O  /\  ( P `  w )  =  (/) ) )  -> 
( ( E. z  e.  A  ( F `  w )  =  (inl
`  z )  /\  (/)  =  (/) )  \/  ( -.  E. z  e.  A  ( F `  w )  =  (inl `  z
)  /\  (/)  =  1o ) ) )
275, 26ecased 1285 . . . . 5  |-  ( (
ph  /\  ( w  e.  O  /\  ( P `  w )  =  (/) ) )  -> 
( E. z  e.  A  ( F `  w )  =  (inl
`  z )  /\  (/)  =  (/) ) )
2827simpld 110 . . . 4  |-  ( (
ph  /\  ( w  e.  O  /\  ( P `  w )  =  (/) ) )  ->  E. z  e.  A  ( F `  w )  =  (inl `  z
) )
29 rexm 3377 . . . 4  |-  ( E. z  e.  A  ( F `  w )  =  (inl `  z
)  ->  E. z 
z  e.  A )
3028, 29syl 14 . . 3  |-  ( (
ph  /\  ( w  e.  O  /\  ( P `  w )  =  (/) ) )  ->  E. z  z  e.  A )
31 eleq1w 2148 . . . 4  |-  ( z  =  x  ->  (
z  e.  A  <->  x  e.  A ) )
3231cbvexv 1843 . . 3  |-  ( E. z  z  e.  A  <->  E. x  x  e.  A
)
3330, 32sylib 120 . 2  |-  ( (
ph  /\  ( w  e.  O  /\  ( P `  w )  =  (/) ) )  ->  E. x  x  e.  A )
341, 33rexlimddv 2493 1  |-  ( ph  ->  E. x  x  e.  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    <-> wb 103    \/ wo 664  DECID wdc 780    = wceq 1289   E.wex 1426    e. wcel 1438   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-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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