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Theorem cantnfsOLD 8008
Description: Elementhood in the set of finitely supported functions from 
B to  A. (Contributed by Mario Carneiro, 25-May-2015.) Obsolete version of cantnfs 7978 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 2451 . . . . . 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 7976 . . . . 5  |-  ( ph  ->  dom  ( A CNF  B
)  =  { g  e.  ( A  ^m  B )  |  ( `' g " ( _V  \  1o ) )  e.  Fin } )
61, 5syl5eq 2504 . . . 4  |-  ( ph  ->  S  =  { g  e.  ( A  ^m  B )  |  ( `' g " ( _V  \  1o ) )  e.  Fin } )
76eleq2d 2521 . . 3  |-  ( ph  ->  ( F  e.  S  <->  F  e.  { g  e.  ( A  ^m  B
)  |  ( `' g " ( _V 
\  1o ) )  e.  Fin } ) )
8 cnveq 5114 . . . . . 6  |-  ( g  =  F  ->  `' g  =  `' F
)
98imaeq1d 5269 . . . . 5  |-  ( g  =  F  ->  ( `' g " ( _V  \  1o ) )  =  ( `' F " ( _V  \  1o ) ) )
109eleq1d 2520 . . . 4  |-  ( g  =  F  ->  (
( `' g "
( _V  \  1o ) )  e.  Fin  <->  ( `' F " ( _V 
\  1o ) )  e.  Fin ) )
1110elrab 3217 . . 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 7330 . . . 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 1370    e. wcel 1758   {crab 2799   _Vcvv 3071    \ cdif 3426   Oncon0 4820   `'ccnv 4940   dom cdm 4941   "cima 4944   -->wf 5515  (class class class)co 6193   1oc1o 7016    ^m cmap 7317   Fincfn 7413   CNF ccnf 7971
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-id 4737  df-po 4742  df-so 4743  df-fr 4780  df-se 4781  df-we 4782  df-suc 4826  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-riota 6154  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-1st 6680  df-2nd 6681  df-supp 6794  df-recs 6935  df-rdg 6969  df-seqom 7006  df-1o 7023  df-map 7319  df-fsupp 7725  df-oi 7828  df-cnf 7972
This theorem is referenced by:  cantnfclOLD  8009  cantnfleOLD  8013  cantnfltOLD  8014  cantnfp1lem1OLD  8016  cantnfp1lem2OLD  8017  cantnfp1lem3OLD  8018  cantnfp1OLD  8019  cantnflem1aOLD  8020  cantnflem1bOLD  8021  cantnflem1cOLD  8022  cantnflem1dOLD  8023  cantnflem1OLD  8024  cantnflem3OLD  8025  cantnfOLD  8027  cnfcomlemOLD  8044  cnfcomOLD  8045  cnfcom2lemOLD  8046  cnfcom3lemOLD  8048  cnfcom3OLD  8049
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