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Theorem cantnfs 8081
Description: Elementhood in the set of finitely supported functions from 
B to  A. (Contributed by Mario Carneiro, 25-May-2015.) (Revised by AV, 28-Jun-2019.)
Hypotheses
Ref Expression
cantnfs.s  |-  S  =  dom  ( A CNF  B
)
cantnfs.a  |-  ( ph  ->  A  e.  On )
cantnfs.b  |-  ( ph  ->  B  e.  On )
Assertion
Ref Expression
cantnfs  |-  ( ph  ->  ( F  e.  S  <->  ( F : B --> A  /\  F finSupp 
(/) ) ) )

Proof of Theorem cantnfs
Dummy variable  g is distinct from all other variables.
StepHypRef Expression
1 cantnfs.s . . . . 5  |-  S  =  dom  ( A CNF  B
)
2 eqid 2467 . . . . . 6  |-  { g  e.  ( A  ^m  B )  |  g finSupp  (/)
}  =  { g  e.  ( A  ^m  B )  |  g finSupp  (/)
}
3 cantnfs.a . . . . . 6  |-  ( ph  ->  A  e.  On )
4 cantnfs.b . . . . . 6  |-  ( ph  ->  B  e.  On )
52, 3, 4cantnfdm 8077 . . . . 5  |-  ( ph  ->  dom  ( A CNF  B
)  =  { g  e.  ( A  ^m  B )  |  g finSupp  (/)
} )
61, 5syl5eq 2520 . . . 4  |-  ( ph  ->  S  =  { g  e.  ( A  ^m  B )  |  g finSupp  (/)
} )
76eleq2d 2537 . . 3  |-  ( ph  ->  ( F  e.  S  <->  F  e.  { g  e.  ( A  ^m  B
)  |  g finSupp  (/) } ) )
8 breq1 4450 . . . 4  |-  ( g  =  F  ->  (
g finSupp  (/)  <->  F finSupp  (/) ) )
98elrab 3261 . . 3  |-  ( F  e.  { g  e.  ( A  ^m  B
)  |  g finSupp  (/) }  <->  ( F  e.  ( A  ^m  B
)  /\  F finSupp  (/) ) )
107, 9syl6bb 261 . 2  |-  ( ph  ->  ( F  e.  S  <->  ( F  e.  ( A  ^m  B )  /\  F finSupp 
(/) ) ) )
11 elmapg 7430 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( F  e.  ( A  ^m  B )  <-> 
F : B --> A ) )
123, 4, 11syl2anc 661 . . 3  |-  ( ph  ->  ( F  e.  ( A  ^m  B )  <-> 
F : B --> A ) )
1312anbi1d 704 . 2  |-  ( ph  ->  ( ( F  e.  ( A  ^m  B
)  /\  F finSupp  (/) )  <->  ( F : B --> A  /\  F finSupp  (/) ) ) )
1410, 13bitrd 253 1  |-  ( ph  ->  ( F  e.  S  <->  ( F : B --> A  /\  F finSupp 
(/) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   {crab 2818   (/)c0 3785   class class class wbr 4447   Oncon0 4878   dom cdm 4999   -->wf 5582  (class class class)co 6282    ^m cmap 7417   finSupp cfsupp 7825   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-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-ov 6285  df-oprab 6286  df-mpt2 6287  df-recs 7039  df-rdg 7073  df-seqom 7110  df-map 7419  df-cnf 8075
This theorem is referenced by:  cantnfcl  8082  cantnfle  8086  cantnflt  8087  cantnff  8089  cantnf0  8090  cantnfrescl  8091  cantnfp1lem1  8093  cantnfp1lem2  8094  cantnfp1lem3  8095  cantnfp1  8096  oemapvali  8099  cantnflem1a  8100  cantnflem1b  8101  cantnflem1c  8102  cantnflem1d  8103  cantnflem1  8104  cantnflem3  8106  cantnf  8108  cnfcomlem  8139  cnfcom  8140  cnfcom2lem  8141  cnfcom3lem  8143  cnfcom3  8144
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