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Theorem pwf1oexmid 13365
Description: An exercise related to  N copies of a singleton and the power set of a singleton (where the latter can also be thought of as representing truth values). Posed as an exercise by Martin Escardo online. (Contributed by Jim Kingdon, 3-Sep-2023.)
Hypothesis
Ref Expression
pwle2.t  |-  T  = 
U_ x  e.  N  ( { x }  X.  1o )
Assertion
Ref Expression
pwf1oexmid  |-  ( ( N  e.  om  /\  G : T -1-1-> ~P 1o )  ->  ( ran  G  =  ~P 1o  <->  ( N  =  2o  /\ EXMID ) ) )
Distinct variable group:    x, N
Allowed substitution hints:    T( x)    G( x)

Proof of Theorem pwf1oexmid
StepHypRef Expression
1 pwle2.t . . . . . 6  |-  T  = 
U_ x  e.  N  ( { x }  X.  1o )
21pwle2 13364 . . . . 5  |-  ( ( N  e.  om  /\  G : T -1-1-> ~P 1o )  ->  N  C_  2o )
32adantr 274 . . . 4  |-  ( ( ( N  e.  om  /\  G : T -1-1-> ~P 1o )  /\  ran  G  =  ~P 1o )  ->  N  C_  2o )
4 pw1dom2 13359 . . . . . 6  |-  2o  ~<_  ~P 1o
5 iunxpconst 4606 . . . . . . . . . . . 12  |-  U_ x  e.  N  ( {
x }  X.  1o )  =  ( N  X.  1o )
6 df1o2 6333 . . . . . . . . . . . . 13  |-  1o  =  { (/) }
76xpeq2i 4567 . . . . . . . . . . . 12  |-  ( N  X.  1o )  =  ( N  X.  { (/)
} )
81, 5, 73eqtri 2165 . . . . . . . . . . 11  |-  T  =  ( N  X.  { (/)
} )
9 peano1 4515 . . . . . . . . . . . 12  |-  (/)  e.  om
10 xpsneng 6723 . . . . . . . . . . . 12  |-  ( ( N  e.  om  /\  (/) 
e.  om )  ->  ( N  X.  { (/) } ) 
~~  N )
119, 10mpan2 422 . . . . . . . . . . 11  |-  ( N  e.  om  ->  ( N  X.  { (/) } ) 
~~  N )
128, 11eqbrtrid 3970 . . . . . . . . . 10  |-  ( N  e.  om  ->  T  ~~  N )
1312ad2antrr 480 . . . . . . . . 9  |-  ( ( ( N  e.  om  /\  G : T -1-1-> ~P 1o )  /\  ran  G  =  ~P 1o )  ->  T  ~~  N )
1413ensymd 6684 . . . . . . . 8  |-  ( ( ( N  e.  om  /\  G : T -1-1-> ~P 1o )  /\  ran  G  =  ~P 1o )  ->  N  ~~  T )
15 relen 6645 . . . . . . . . . 10  |-  Rel  ~~
16 brrelex1 4585 . . . . . . . . . 10  |-  ( ( Rel  ~~  /\  T  ~~  N )  ->  T  e.  _V )
1715, 13, 16sylancr 411 . . . . . . . . 9  |-  ( ( ( N  e.  om  /\  G : T -1-1-> ~P 1o )  /\  ran  G  =  ~P 1o )  ->  T  e.  _V )
18 simplr 520 . . . . . . . . . 10  |-  ( ( ( N  e.  om  /\  G : T -1-1-> ~P 1o )  /\  ran  G  =  ~P 1o )  ->  G : T -1-1-> ~P 1o )
19 simpr 109 . . . . . . . . . 10  |-  ( ( ( N  e.  om  /\  G : T -1-1-> ~P 1o )  /\  ran  G  =  ~P 1o )  ->  ran  G  =  ~P 1o )
20 dff1o5 5383 . . . . . . . . . 10  |-  ( G : T -1-1-onto-> ~P 1o  <->  ( G : T -1-1-> ~P 1o  /\  ran  G  =  ~P 1o ) )
2118, 19, 20sylanbrc 414 . . . . . . . . 9  |-  ( ( ( N  e.  om  /\  G : T -1-1-> ~P 1o )  /\  ran  G  =  ~P 1o )  ->  G : T -1-1-onto-> ~P 1o )
22 f1oeng 6658 . . . . . . . . 9  |-  ( ( T  e.  _V  /\  G : T -1-1-onto-> ~P 1o )  ->  T  ~~  ~P 1o )
2317, 21, 22syl2anc 409 . . . . . . . 8  |-  ( ( ( N  e.  om  /\  G : T -1-1-> ~P 1o )  /\  ran  G  =  ~P 1o )  ->  T  ~~  ~P 1o )
24 entr 6685 . . . . . . . 8  |-  ( ( N  ~~  T  /\  T  ~~  ~P 1o )  ->  N  ~~  ~P 1o )
2514, 23, 24syl2anc 409 . . . . . . 7  |-  ( ( ( N  e.  om  /\  G : T -1-1-> ~P 1o )  /\  ran  G  =  ~P 1o )  ->  N  ~~  ~P 1o )
2625ensymd 6684 . . . . . 6  |-  ( ( ( N  e.  om  /\  G : T -1-1-> ~P 1o )  /\  ran  G  =  ~P 1o )  ->  ~P 1o  ~~  N )
27 domentr 6692 . . . . . 6  |-  ( ( 2o  ~<_  ~P 1o  /\  ~P 1o  ~~  N )  ->  2o 
~<_  N )
284, 26, 27sylancr 411 . . . . 5  |-  ( ( ( N  e.  om  /\  G : T -1-1-> ~P 1o )  /\  ran  G  =  ~P 1o )  ->  2o 
~<_  N )
29 2onn 6424 . . . . . . 7  |-  2o  e.  om
30 nndomo 6765 . . . . . . 7  |-  ( ( 2o  e.  om  /\  N  e.  om )  ->  ( 2o  ~<_  N  <->  2o  C_  N
) )
3129, 30mpan 421 . . . . . 6  |-  ( N  e.  om  ->  ( 2o 
~<_  N  <->  2o  C_  N ) )
3231ad2antrr 480 . . . . 5  |-  ( ( ( N  e.  om  /\  G : T -1-1-> ~P 1o )  /\  ran  G  =  ~P 1o )  -> 
( 2o  ~<_  N  <->  2o  C_  N
) )
3328, 32mpbid 146 . . . 4  |-  ( ( ( N  e.  om  /\  G : T -1-1-> ~P 1o )  /\  ran  G  =  ~P 1o )  ->  2o  C_  N )
343, 33eqssd 3118 . . 3  |-  ( ( ( N  e.  om  /\  G : T -1-1-> ~P 1o )  /\  ran  G  =  ~P 1o )  ->  N  =  2o )
3526, 34breqtrd 3961 . . . 4  |-  ( ( ( N  e.  om  /\  G : T -1-1-> ~P 1o )  /\  ran  G  =  ~P 1o )  ->  ~P 1o  ~~  2o )
36 exmidpw 6809 . . . 4  |-  (EXMID  <->  ~P 1o  ~~  2o )
3735, 36sylibr 133 . . 3  |-  ( ( ( N  e.  om  /\  G : T -1-1-> ~P 1o )  /\  ran  G  =  ~P 1o )  -> EXMID )
3834, 37jca 304 . 2  |-  ( ( ( N  e.  om  /\  G : T -1-1-> ~P 1o )  /\  ran  G  =  ~P 1o )  -> 
( N  =  2o 
/\ EXMID
) )
39 simplr 520 . . . . 5  |-  ( ( ( N  e.  om  /\  G : T -1-1-> ~P 1o )  /\  ( N  =  2o  /\ EXMID ) )  ->  G : T -1-1-> ~P 1o )
4012ad2antrr 480 . . . . . . . 8  |-  ( ( ( N  e.  om  /\  G : T -1-1-> ~P 1o )  /\  ( N  =  2o  /\ EXMID ) )  ->  T  ~~  N )
41 simprl 521 . . . . . . . 8  |-  ( ( ( N  e.  om  /\  G : T -1-1-> ~P 1o )  /\  ( N  =  2o  /\ EXMID ) )  ->  N  =  2o )
4240, 41breqtrd 3961 . . . . . . 7  |-  ( ( ( N  e.  om  /\  G : T -1-1-> ~P 1o )  /\  ( N  =  2o  /\ EXMID ) )  ->  T  ~~  2o )
43 simprr 522 . . . . . . . . 9  |-  ( ( ( N  e.  om  /\  G : T -1-1-> ~P 1o )  /\  ( N  =  2o  /\ EXMID ) )  -> EXMID )
4443, 36sylib 121 . . . . . . . 8  |-  ( ( ( N  e.  om  /\  G : T -1-1-> ~P 1o )  /\  ( N  =  2o  /\ EXMID ) )  ->  ~P 1o  ~~  2o )
4544ensymd 6684 . . . . . . 7  |-  ( ( ( N  e.  om  /\  G : T -1-1-> ~P 1o )  /\  ( N  =  2o  /\ EXMID ) )  ->  2o  ~~  ~P 1o )
46 entr 6685 . . . . . . 7  |-  ( ( T  ~~  2o  /\  2o  ~~  ~P 1o )  ->  T  ~~  ~P 1o )
4742, 45, 46syl2anc 409 . . . . . 6  |-  ( ( ( N  e.  om  /\  G : T -1-1-> ~P 1o )  /\  ( N  =  2o  /\ EXMID ) )  ->  T  ~~  ~P 1o )
48 nnfi 6773 . . . . . . . 8  |-  ( 2o  e.  om  ->  2o  e.  Fin )
4929, 48mp1i 10 . . . . . . 7  |-  ( ( ( N  e.  om  /\  G : T -1-1-> ~P 1o )  /\  ( N  =  2o  /\ EXMID ) )  ->  2o  e.  Fin )
50 enfi 6774 . . . . . . . 8  |-  ( ~P 1o  ~~  2o  ->  ( ~P 1o  e.  Fin  <->  2o  e.  Fin ) )
5144, 50syl 14 . . . . . . 7  |-  ( ( ( N  e.  om  /\  G : T -1-1-> ~P 1o )  /\  ( N  =  2o  /\ EXMID ) )  -> 
( ~P 1o  e.  Fin 
<->  2o  e.  Fin )
)
5249, 51mpbird 166 . . . . . 6  |-  ( ( ( N  e.  om  /\  G : T -1-1-> ~P 1o )  /\  ( N  =  2o  /\ EXMID ) )  ->  ~P 1o  e.  Fin )
53 f1finf1o 6842 . . . . . 6  |-  ( ( T  ~~  ~P 1o  /\ 
~P 1o  e.  Fin )  ->  ( G : T -1-1-> ~P 1o  <->  G : T
-1-1-onto-> ~P 1o ) )
5447, 52, 53syl2anc 409 . . . . 5  |-  ( ( ( N  e.  om  /\  G : T -1-1-> ~P 1o )  /\  ( N  =  2o  /\ EXMID ) )  -> 
( G : T -1-1-> ~P 1o  <->  G : T -1-1-onto-> ~P 1o ) )
5539, 54mpbid 146 . . . 4  |-  ( ( ( N  e.  om  /\  G : T -1-1-> ~P 1o )  /\  ( N  =  2o  /\ EXMID ) )  ->  G : T -1-1-onto-> ~P 1o )
5655, 20sylib 121 . . 3  |-  ( ( ( N  e.  om  /\  G : T -1-1-> ~P 1o )  /\  ( N  =  2o  /\ EXMID ) )  -> 
( G : T -1-1-> ~P 1o  /\  ran  G  =  ~P 1o ) )
5756simprd 113 . 2  |-  ( ( ( N  e.  om  /\  G : T -1-1-> ~P 1o )  /\  ( N  =  2o  /\ EXMID ) )  ->  ran  G  =  ~P 1o )
5838, 57impbida 586 1  |-  ( ( N  e.  om  /\  G : T -1-1-> ~P 1o )  ->  ( ran  G  =  ~P 1o  <->  ( N  =  2o  /\ EXMID ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1332    e. wcel 1481   _Vcvv 2689    C_ wss 3075   (/)c0 3367   ~Pcpw 3514   {csn 3531   U_ciun 3820   class class class wbr 3936  EXMIDwem 4125   omcom 4511    X. cxp 4544   ran crn 4547   Rel wrel 4551   -1-1->wf1 5127   -1-1-onto->wf1o 5129   1oc1o 6313   2oc2o 6314    ~~ cen 6639    ~<_ cdom 6640   Fincfn 6641
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-exmid 4126  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-1o 6320  df-2o 6321  df-er 6436  df-en 6642  df-dom 6643  df-fin 6644
This theorem is referenced by: (None)
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