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Theorem cantnfs 7367
Description: Elementhood in the set of finitely supported functions from 
B to  A. (Contributed by Mario Carneiro, 25-May-2015.)
Hypotheses
Ref Expression
cantnfs.1  |-  S  =  dom  ( A CNF  B
)
cantnfs.2  |-  ( ph  ->  A  e.  On )
cantnfs.3  |-  ( ph  ->  B  e.  On )
Assertion
Ref Expression
cantnfs  |-  ( ph  ->  ( F  e.  S  <->  ( F : B --> A  /\  ( `' F " ( _V 
\  1o ) )  e.  Fin ) ) )

Proof of Theorem cantnfs
Dummy variable  g is distinct from all other variables.
StepHypRef Expression
1 cantnfs.1 . . . . 5  |-  S  =  dom  ( A CNF  B
)
2 eqid 2283 . . . . . 6  |-  { g  e.  ( A  ^m  B )  |  ( `' g " ( _V  \  1o ) )  e.  Fin }  =  { g  e.  ( A  ^m  B )  |  ( `' g
" ( _V  \  1o ) )  e.  Fin }
3 cantnfs.2 . . . . . 6  |-  ( ph  ->  A  e.  On )
4 cantnfs.3 . . . . . 6  |-  ( ph  ->  B  e.  On )
52, 3, 4cantnfdm 7365 . . . . 5  |-  ( ph  ->  dom  ( A CNF  B
)  =  { g  e.  ( A  ^m  B )  |  ( `' g " ( _V  \  1o ) )  e.  Fin } )
61, 5syl5eq 2327 . . . 4  |-  ( ph  ->  S  =  { g  e.  ( A  ^m  B )  |  ( `' g " ( _V  \  1o ) )  e.  Fin } )
76eleq2d 2350 . . 3  |-  ( ph  ->  ( F  e.  S  <->  F  e.  { g  e.  ( A  ^m  B
)  |  ( `' g " ( _V 
\  1o ) )  e.  Fin } ) )
8 cnveq 4855 . . . . . 6  |-  ( g  =  F  ->  `' g  =  `' F
)
98imaeq1d 5011 . . . . 5  |-  ( g  =  F  ->  ( `' g " ( _V  \  1o ) )  =  ( `' F " ( _V  \  1o ) ) )
109eleq1d 2349 . . . 4  |-  ( g  =  F  ->  (
( `' g "
( _V  \  1o ) )  e.  Fin  <->  ( `' F " ( _V 
\  1o ) )  e.  Fin ) )
1110elrab 2923 . . 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 252 . 2  |-  ( ph  ->  ( F  e.  S  <->  ( F  e.  ( A  ^m  B )  /\  ( `' F " ( _V 
\  1o ) )  e.  Fin ) ) )
13 elmapg 6785 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( F  e.  ( A  ^m  B )  <-> 
F : B --> A ) )
143, 4, 13syl2anc 642 . . 3  |-  ( ph  ->  ( F  e.  ( A  ^m  B )  <-> 
F : B --> A ) )
1514anbi1d 685 . 2  |-  ( ph  ->  ( ( F  e.  ( A  ^m  B
)  /\  ( `' F " ( _V  \  1o ) )  e.  Fin ) 
<->  ( F : B --> A  /\  ( `' F " ( _V  \  1o ) )  e.  Fin ) ) )
1612, 15bitrd 244 1  |-  ( ph  ->  ( F  e.  S  <->  ( F : B --> A  /\  ( `' F " ( _V 
\  1o ) )  e.  Fin ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   {crab 2547   _Vcvv 2788    \ cdif 3149   Oncon0 4392   `'ccnv 4688   dom cdm 4689   "cima 4692   -->wf 5251  (class class class)co 5858   1oc1o 6472    ^m cmap 6772   Fincfn 6863   CNF ccnf 7362
This theorem is referenced by:  cantnfcl  7368  cantnfle  7372  cantnflt  7373  cantnff  7375  cantnf0  7376  cantnfrescl  7378  cantnfp1lem1  7380  cantnfp1lem2  7381  cantnfp1lem3  7382  cantnfp1  7383  oemapvali  7386  cantnflem1a  7387  cantnflem1b  7388  cantnflem1c  7389  cantnflem1d  7390  cantnflem1  7391  cantnflem3  7393  cantnf  7395  cnfcomlem  7402  cnfcom  7403  cnfcom2lem  7404  cnfcom3lem  7406  cnfcom3  7407
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-riota 6304  df-recs 6388  df-rdg 6423  df-seqom 6460  df-map 6774  df-oi 7225  df-cnf 7363
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