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Theorem nninfomnilem 13387
Description: Lemma for nninfomni 13388. (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 13378 . . 3  |-  e.  _V
2 isomnimap 7016 . . 3  |-  (  e.  _V  ->  ( 
e. Omni 
<-> 
A. r  e.  ( 2o  ^m ) ( E. p  e.  ( r `  p )  =  (/)  \/  A. p  e.  ( r `  p )  =  1o ) ) )
31, 2ax-mp 5 . 2  |-  (  e. Omni  <->  A. r  e.  ( 2o  ^m ) ( E. p  e.  ( r `  p )  =  (/)  \/  A. p  e.  ( r `  p )  =  1o ) )
4 elmapi 6571 . . . . . 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 13382 . . . . . . 7  |-  E :
( 2o  ^m ) -->
76ffvelrni 5561 . . . . . 6  |-  ( r  e.  ( 2o  ^m )  -> 
( E `  r
)  e. )
84, 7ffvelrnd 5563 . . . . 5  |-  ( r  e.  ( 2o  ^m )  -> 
( r `  ( E `  r )
)  e.  2o )
9 df2o3 6334 . . . . 5  |-  2o  =  { (/) ,  1o }
108, 9eleqtrdi 2233 . . . 4  |-  ( r  e.  ( 2o  ^m )  -> 
( r `  ( E `  r )
)  e.  { (/) ,  1o } )
11 elpri 3554 . . . 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 5428 . . . . . . . 8  |-  ( p  =  ( E `  r )  ->  (
r `  p )  =  ( r `  ( E `  r ) ) )
1413eqeq1d 2149 . . . . . . 7  |-  ( p  =  ( E `  r )  ->  (
( r `  p
)  =  (/)  <->  ( r `  ( E `  r
) )  =  (/) ) )
1514rspcev 2792 . . . . . 6  |-  ( ( ( E `  r
)  e.  /\  ( r `  ( E `  r ) )  =  (/) )  ->  E. p  e.  ( r `  p
)  =  (/) )
1615ex 114 . . . . 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 108 . . . . . 6  |-  ( ( r  e.  ( 2o 
^m )  /\  ( r `  ( E `  r ) )  =  1o )  ->  r  e.  ( 2o  ^m ) )
19 simpr 109 . . . . . 6  |-  ( ( r  e.  ( 2o 
^m )  /\  ( r `  ( E `  r ) )  =  1o )  ->  ( r `  ( E `  r ) )  =  1o )
205, 18, 19nninfsel 13386 . . . . 5  |-  ( ( r  e.  ( 2o 
^m )  /\  ( r `  ( E `  r ) )  =  1o )  ->  A. p  e.  ( r `  p
)  =  1o )
2120ex 114 . . . 4  |-  ( r  e.  ( 2o  ^m )  -> 
( ( r `  ( E `  r ) )  =  1o  ->  A. p  e.  ( r `  p
)  =  1o ) )
2217, 21orim12d 776 . . 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 2493 1  |-  e. Omni
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    \/ wo 698    = wceq 1332    e. wcel 1481   A.wral 2417   E.wrex 2418   _Vcvv 2689   (/)c0 3367   ifcif 3478   {cpr 3532    |-> cmpt 3996   suc csuc 4294   omcom 4511   ` cfv 5130  (class class class)co 5781   1oc1o 6313   2oc2o 6314    ^m cmap 6549  Omnicomni 7011  ℕxnninf 7012
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-coll 4050  ax-sep 4053  ax-nul 4061  ax-pow 4105  ax-pr 4138  ax-un 4362  ax-setind 4459  ax-iinf 4509
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  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-reu 2424  df-rab 2426  df-v 2691  df-sbc 2913  df-csb 3007  df-dif 3077  df-un 3079  df-in 3081  df-ss 3088  df-nul 3368  df-if 3479  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-int 3779  df-iun 3822  df-br 3937  df-opab 3997  df-mpt 3998  df-tr 4034  df-id 4222  df-iord 4295  df-on 4297  df-suc 4300  df-iom 4512  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-rn 4557  df-res 4558  df-ima 4559  df-iota 5095  df-fun 5132  df-fn 5133  df-f 5134  df-f1 5135  df-fo 5136  df-f1o 5137  df-fv 5138  df-ov 5784  df-oprab 5785  df-mpo 5786  df-1o 6320  df-2o 6321  df-map 6551  df-omni 7013  df-nninf 7014
This theorem is referenced by:  nninfomni  13388
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