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Theorem nninfomnilem 11339
Description: Lemma for nninfomni 11340. (Contributed by Jim Kingdon, 10-Aug-2022.)
Hypothesis
Ref Expression
nninfsel.e  |-  E  =  ( q  e.  ( 2o  ^m )  |->  ( n  e. 
om  |->  if ( A. k  e.  suc  n ( q `  ( i  e.  om  |->  if ( i  e.  k ,  1o ,  (/) ) ) )  =  1o ,  1o ,  (/) ) ) )
Assertion
Ref Expression
nninfomnilem  |-  e. Omni
Distinct variable groups:    i, E, k, n    i, q, k, n
Allowed substitution hint:    E( q)

Proof of Theorem nninfomnilem
Dummy variables  p  r are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nninfex 11330 . . 3  |-  e.  _V
2 isomnimap 6729 . . 3  |-  (  e.  _V  ->  ( 
e. Omni 
<-> 
A. r  e.  ( 2o  ^m ) ( E. p  e.  ( r `  p )  =  (/)  \/  A. p  e.  ( r `  p )  =  1o ) ) )
31, 2ax-mp 7 . 2  |-  (  e. Omni  <->  A. r  e.  ( 2o  ^m ) ( E. p  e.  ( r `  p )  =  (/)  \/  A. p  e.  ( r `  p )  =  1o ) )
4 elmapi 6372 . . . . . 6  |-  ( r  e.  ( 2o  ^m )  -> 
r : --> 2o )
5 nninfsel.e . . . . . . . 8  |-  E  =  ( q  e.  ( 2o  ^m )  |->  ( n  e. 
om  |->  if ( A. k  e.  suc  n ( q `  ( i  e.  om  |->  if ( i  e.  k ,  1o ,  (/) ) ) )  =  1o ,  1o ,  (/) ) ) )
65nninfself 11334 . . . . . . 7  |-  E :
( 2o  ^m ) -->
76ffvelrni 5389 . . . . . 6  |-  ( r  e.  ( 2o  ^m )  -> 
( E `  r
)  e. )
84, 7ffvelrnd 5391 . . . . 5  |-  ( r  e.  ( 2o  ^m )  -> 
( r `  ( E `  r )
)  e.  2o )
9 df2o3 6142 . . . . 5  |-  2o  =  { (/) ,  1o }
108, 9syl6eleq 2177 . . . 4  |-  ( r  e.  ( 2o  ^m )  -> 
( r `  ( E `  r )
)  e.  { (/) ,  1o } )
11 elpri 3453 . . . 4  |-  ( ( r `  ( E `
 r ) )  e.  { (/) ,  1o }  ->  ( ( r `
 ( E `  r ) )  =  (/)  \/  ( r `  ( E `  r ) )  =  1o ) )
1210, 11syl 14 . . 3  |-  ( r  e.  ( 2o  ^m )  -> 
( ( r `  ( E `  r ) )  =  (/)  \/  (
r `  ( E `  r ) )  =  1o ) )
13 fveq2 5261 . . . . . . . 8  |-  ( p  =  ( E `  r )  ->  (
r `  p )  =  ( r `  ( E `  r ) ) )
1413eqeq1d 2093 . . . . . . 7  |-  ( p  =  ( E `  r )  ->  (
( r `  p
)  =  (/)  <->  ( r `  ( E `  r
) )  =  (/) ) )
1514rspcev 2715 . . . . . 6  |-  ( ( ( E `  r
)  e.  /\  ( r `  ( E `  r ) )  =  (/) )  ->  E. p  e.  ( r `  p
)  =  (/) )
1615ex 113 . . . . 5  |-  ( ( E `  r )  e.  ->  ( ( r `  ( E `  r ) )  =  (/)  ->  E. p  e.  ( r `  p )  =  (/) ) )
177, 16syl 14 . . . 4  |-  ( r  e.  ( 2o  ^m )  -> 
( ( r `  ( E `  r ) )  =  (/)  ->  E. p  e.  ( r `  p )  =  (/) ) )
18 simpl 107 . . . . . 6  |-  ( ( r  e.  ( 2o 
^m )  /\  ( r `  ( E `  r ) )  =  1o )  ->  r  e.  ( 2o  ^m ) )
19 simpr 108 . . . . . 6  |-  ( ( r  e.  ( 2o 
^m )  /\  ( r `  ( E `  r ) )  =  1o )  ->  ( r `  ( E `  r ) )  =  1o )
205, 18, 19nninfsel 11338 . . . . 5  |-  ( ( r  e.  ( 2o 
^m )  /\  ( r `  ( E `  r ) )  =  1o )  ->  A. p  e.  ( r `  p
)  =  1o )
2120ex 113 . . . 4  |-  ( r  e.  ( 2o  ^m )  -> 
( ( r `  ( E `  r ) )  =  1o  ->  A. p  e.  ( r `  p
)  =  1o ) )
2217, 21orim12d 733 . . 3  |-  ( r  e.  ( 2o  ^m )  -> 
( ( ( r `
 ( E `  r ) )  =  (/)  \/  ( r `  ( E `  r ) )  =  1o )  ->  ( E. p  e.  ( r `  p )  =  (/)  \/  A. p  e.  ( r `  p )  =  1o ) ) )
2312, 22mpd 13 . 2  |-  ( r  e.  ( 2o  ^m )  -> 
( E. p  e.  (
r `  p )  =  (/)  \/  A. p  e.  ( r `  p )  =  1o ) )
243, 23mprgbir 2429 1  |-  e. Omni
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    \/ wo 662    = wceq 1287    e. wcel 1436   A.wral 2355   E.wrex 2356   _Vcvv 2615   (/)c0 3275   ifcif 3379   {cpr 3431    |-> cmpt 3873   suc csuc 4164   omcom 4376   ` cfv 4977  (class class class)co 5606   1oc1o 6121   2oc2o 6122    ^m cmap 6350  Omnicomni 6724  ℕxnninf 6725
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 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-coll 3927  ax-sep 3930  ax-nul 3938  ax-pow 3982  ax-pr 4008  ax-un 4232  ax-setind 4324  ax-iinf 4374
This theorem depends on definitions:  df-bi 115  df-dc 779  df-3or 923  df-3an 924  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ne 2252  df-ral 2360  df-rex 2361  df-reu 2362  df-rab 2364  df-v 2617  df-sbc 2830  df-csb 2923  df-dif 2990  df-un 2992  df-in 2994  df-ss 3001  df-nul 3276  df-if 3380  df-pw 3416  df-sn 3436  df-pr 3437  df-op 3439  df-uni 3636  df-int 3671  df-iun 3714  df-br 3820  df-opab 3874  df-mpt 3875  df-tr 3910  df-id 4092  df-iord 4165  df-on 4167  df-suc 4170  df-iom 4377  df-xp 4415  df-rel 4416  df-cnv 4417  df-co 4418  df-dm 4419  df-rn 4420  df-res 4421  df-ima 4422  df-iota 4942  df-fun 4979  df-fn 4980  df-f 4981  df-f1 4982  df-fo 4983  df-f1o 4984  df-fv 4985  df-ov 5609  df-oprab 5610  df-mpt2 5611  df-1o 6128  df-2o 6129  df-map 6352  df-omni 6726  df-nninf 6727
This theorem is referenced by:  nninfomni  11340
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