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Theorem cantnfsOLD 8111
Description: Elementhood in the set of finitely supported functions from 
B to  A. (Contributed by Mario Carneiro, 25-May-2015.) Obsolete version of cantnfs 8081 as of 28-Jun-2019. (New usage is discouraged.)
Hypotheses
Ref Expression
cantnfsOLD.1  |-  S  =  dom  ( A CNF  B
)
cantnfsOLD.2  |-  ( ph  ->  A  e.  On )
cantnfsOLD.3  |-  ( ph  ->  B  e.  On )
Assertion
Ref Expression
cantnfsOLD  |-  ( ph  ->  ( F  e.  S  <->  ( F : B --> A  /\  ( `' F " ( _V 
\  1o ) )  e.  Fin ) ) )

Proof of Theorem cantnfsOLD
Dummy variable  g is distinct from all other variables.
StepHypRef Expression
1 cantnfsOLD.1 . . . . 5  |-  S  =  dom  ( A CNF  B
)
2 eqid 2467 . . . . . 6  |-  { g  e.  ( A  ^m  B )  |  ( `' g " ( _V  \  1o ) )  e.  Fin }  =  { g  e.  ( A  ^m  B )  |  ( `' g
" ( _V  \  1o ) )  e.  Fin }
3 cantnfsOLD.2 . . . . . 6  |-  ( ph  ->  A  e.  On )
4 cantnfsOLD.3 . . . . . 6  |-  ( ph  ->  B  e.  On )
52, 3, 4cantnfdmOLD 8079 . . . . 5  |-  ( ph  ->  dom  ( A CNF  B
)  =  { g  e.  ( A  ^m  B )  |  ( `' g " ( _V  \  1o ) )  e.  Fin } )
61, 5syl5eq 2520 . . . 4  |-  ( ph  ->  S  =  { g  e.  ( A  ^m  B )  |  ( `' g " ( _V  \  1o ) )  e.  Fin } )
76eleq2d 2537 . . 3  |-  ( ph  ->  ( F  e.  S  <->  F  e.  { g  e.  ( A  ^m  B
)  |  ( `' g " ( _V 
\  1o ) )  e.  Fin } ) )
8 cnveq 5174 . . . . . 6  |-  ( g  =  F  ->  `' g  =  `' F
)
98imaeq1d 5334 . . . . 5  |-  ( g  =  F  ->  ( `' g " ( _V  \  1o ) )  =  ( `' F " ( _V  \  1o ) ) )
109eleq1d 2536 . . . 4  |-  ( g  =  F  ->  (
( `' g "
( _V  \  1o ) )  e.  Fin  <->  ( `' F " ( _V 
\  1o ) )  e.  Fin ) )
1110elrab 3261 . . 3  |-  ( F  e.  { g  e.  ( A  ^m  B
)  |  ( `' g " ( _V 
\  1o ) )  e.  Fin }  <->  ( F  e.  ( A  ^m  B
)  /\  ( `' F " ( _V  \  1o ) )  e.  Fin ) )
127, 11syl6bb 261 . 2  |-  ( ph  ->  ( F  e.  S  <->  ( F  e.  ( A  ^m  B )  /\  ( `' F " ( _V 
\  1o ) )  e.  Fin ) ) )
13 elmapg 7430 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( F  e.  ( A  ^m  B )  <-> 
F : B --> A ) )
143, 4, 13syl2anc 661 . . 3  |-  ( ph  ->  ( F  e.  ( A  ^m  B )  <-> 
F : B --> A ) )
1514anbi1d 704 . 2  |-  ( ph  ->  ( ( F  e.  ( A  ^m  B
)  /\  ( `' F " ( _V  \  1o ) )  e.  Fin ) 
<->  ( F : B --> A  /\  ( `' F " ( _V  \  1o ) )  e.  Fin ) ) )
1612, 15bitrd 253 1  |-  ( ph  ->  ( F  e.  S  <->  ( F : B --> A  /\  ( `' F " ( _V 
\  1o ) )  e.  Fin ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   {crab 2818   _Vcvv 3113    \ cdif 3473   Oncon0 4878   `'ccnv 4998   dom cdm 4999   "cima 5002   -->wf 5582  (class class class)co 6282   1oc1o 7120    ^m cmap 7417   Fincfn 7513   CNF ccnf 8074
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-1st 6781  df-2nd 6782  df-supp 6899  df-recs 7039  df-rdg 7073  df-seqom 7110  df-1o 7127  df-map 7419  df-fsupp 7826  df-oi 7931  df-cnf 8075
This theorem is referenced by:  cantnfclOLD  8112  cantnfleOLD  8116  cantnfltOLD  8117  cantnfp1lem1OLD  8119  cantnfp1lem2OLD  8120  cantnfp1lem3OLD  8121  cantnfp1OLD  8122  cantnflem1aOLD  8123  cantnflem1bOLD  8124  cantnflem1cOLD  8125  cantnflem1dOLD  8126  cantnflem1OLD  8127  cantnflem3OLD  8128  cantnfOLD  8130  cnfcomlemOLD  8147  cnfcomOLD  8148  cnfcom2lemOLD  8149  cnfcom3lemOLD  8151  cnfcom3OLD  8152
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