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Theorem cantnfsOLD 8132
Description: Elementhood in the set of finitely supported functions from 
B to  A. (Contributed by Mario Carneiro, 25-May-2015.) Obsolete version of cantnfs 8102 as of 28-Jun-2019. (New usage is discouraged.) (Proof modification 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 2457 . . . . . 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 8100 . . . . 5  |-  ( ph  ->  dom  ( A CNF  B
)  =  { g  e.  ( A  ^m  B )  |  ( `' g " ( _V  \  1o ) )  e.  Fin } )
61, 5syl5eq 2510 . . . 4  |-  ( ph  ->  S  =  { g  e.  ( A  ^m  B )  |  ( `' g " ( _V  \  1o ) )  e.  Fin } )
76eleq2d 2527 . . 3  |-  ( ph  ->  ( F  e.  S  <->  F  e.  { g  e.  ( A  ^m  B
)  |  ( `' g " ( _V 
\  1o ) )  e.  Fin } ) )
8 cnveq 5186 . . . . . 6  |-  ( g  =  F  ->  `' g  =  `' F
)
98imaeq1d 5346 . . . . 5  |-  ( g  =  F  ->  ( `' g " ( _V  \  1o ) )  =  ( `' F " ( _V  \  1o ) ) )
109eleq1d 2526 . . . 4  |-  ( g  =  F  ->  (
( `' g "
( _V  \  1o ) )  e.  Fin  <->  ( `' F " ( _V 
\  1o ) )  e.  Fin ) )
1110elrab 3257 . . 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 7451 . . . 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 1395    e. wcel 1819   {crab 2811   _Vcvv 3109    \ cdif 3468   Oncon0 4887   `'ccnv 5007   dom cdm 5008   "cima 5011   -->wf 5590  (class class class)co 6296   1oc1o 7141    ^m cmap 7438   Fincfn 7535   CNF ccnf 8095
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6799  df-2nd 6800  df-supp 6918  df-recs 7060  df-rdg 7094  df-seqom 7131  df-1o 7148  df-map 7440  df-fsupp 7848  df-oi 7953  df-cnf 8096
This theorem is referenced by:  cantnfclOLD  8133  cantnfleOLD  8137  cantnfltOLD  8138  cantnfp1lem1OLD  8140  cantnfp1lem2OLD  8141  cantnfp1lem3OLD  8142  cantnfp1OLD  8143  cantnflem1aOLD  8144  cantnflem1bOLD  8145  cantnflem1cOLD  8146  cantnflem1dOLD  8147  cantnflem1OLD  8148  cantnflem3OLD  8149  cantnfOLD  8151  cnfcomlemOLD  8168  cnfcomOLD  8169  cnfcom2lemOLD  8170  cnfcom3lemOLD  8172  cnfcom3OLD  8173
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