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Theorem nninfomnilem 14416
Description: Lemma for nninfomni 14417. (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 7114 . . 3  |-  e.  _V
2 isomnimap 7129 . . 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 6664 . . . . . 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 14411 . . . . . . 7  |-  E :
( 2o  ^m ) -->
76ffvelcdmi 5646 . . . . . 6  |-  ( r  e.  ( 2o  ^m )  -> 
( E `  r
)  e. )
84, 7ffvelcdmd 5648 . . . . 5  |-  ( r  e.  ( 2o  ^m )  -> 
( r `  ( E `  r )
)  e.  2o )
9 df2o3 6425 . . . . 5  |-  2o  =  { (/) ,  1o }
108, 9eleqtrdi 2270 . . . 4  |-  ( r  e.  ( 2o  ^m )  -> 
( r `  ( E `  r )
)  e.  { (/) ,  1o } )
11 elpri 3614 . . . 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 fveqeq2 5520 . . . . . . 7  |-  ( p  =  ( E `  r )  ->  (
( r `  p
)  =  (/)  <->  ( r `  ( E `  r
) )  =  (/) ) )
1413rspcev 2841 . . . . . 6  |-  ( ( ( E `  r
)  e.  /\  ( r `  ( E `  r ) )  =  (/) )  ->  E. p  e.  ( r `  p
)  =  (/) )
1514ex 115 . . . . 5  |-  ( ( E `  r )  e.  ->  ( ( r `  ( E `  r ) )  =  (/)  ->  E. p  e.  ( r `  p )  =  (/) ) )
167, 15syl 14 . . . 4  |-  ( r  e.  ( 2o  ^m )  -> 
( ( r `  ( E `  r ) )  =  (/)  ->  E. p  e.  ( r `  p )  =  (/) ) )
17 simpl 109 . . . . . 6  |-  ( ( r  e.  ( 2o 
^m )  /\  ( r `  ( E `  r ) )  =  1o )  ->  r  e.  ( 2o  ^m ) )
18 simpr 110 . . . . . 6  |-  ( ( r  e.  ( 2o 
^m )  /\  ( r `  ( E `  r ) )  =  1o )  ->  ( r `  ( E `  r ) )  =  1o )
195, 17, 18nninfsel 14415 . . . . 5  |-  ( ( r  e.  ( 2o 
^m )  /\  ( r `  ( E `  r ) )  =  1o )  ->  A. p  e.  ( r `  p
)  =  1o )
2019ex 115 . . . 4  |-  ( r  e.  ( 2o  ^m )  -> 
( ( r `  ( E `  r ) )  =  1o  ->  A. p  e.  ( r `  p
)  =  1o ) )
2116, 20orim12d 786 . . 3  |-  ( r  e.  ( 2o  ^m )  -> 
( ( ( r `
 ( E `  r ) )  =  (/)  \/  ( r `  ( E `  r ) )  =  1o )  ->  ( E. p  e.  ( r `  p )  =  (/)  \/  A. p  e.  ( r `  p )  =  1o ) ) )
2212, 21mpd 13 . 2  |-  ( r  e.  ( 2o  ^m )  -> 
( E. p  e.  (
r `  p )  =  (/)  \/  A. p  e.  ( r `  p )  =  1o ) )
233, 22mprgbir 2535 1  |-  e. Omni
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    \/ wo 708    = wceq 1353    e. wcel 2148   A.wral 2455   E.wrex 2456   _Vcvv 2737   (/)c0 3422   ifcif 3534   {cpr 3592    |-> cmpt 4061   suc csuc 4362   omcom 4586   ` cfv 5212  (class class class)co 5869   1oc1o 6404   2oc2o 6405    ^m cmap 6642  ℕxnninf 7112  Omnicomni 7126
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4290  df-iord 4363  df-on 4365  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1o 6411  df-2o 6412  df-map 6644  df-nninf 7113  df-omni 7127
This theorem is referenced by:  nninfomni  14417
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