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Theorem cantnfs 8102
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 2457 . . . . . 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 8098 . . . . 5  |-  ( ph  ->  dom  ( A CNF  B
)  =  { g  e.  ( A  ^m  B )  |  g finSupp  (/)
} )
61, 5syl5eq 2510 . . . 4  |-  ( ph  ->  S  =  { g  e.  ( A  ^m  B )  |  g finSupp  (/)
} )
76eleq2d 2527 . . 3  |-  ( ph  ->  ( F  e.  S  <->  F  e.  { g  e.  ( A  ^m  B
)  |  g finSupp  (/) } ) )
8 breq1 4459 . . . 4  |-  ( g  =  F  ->  (
g finSupp  (/)  <->  F finSupp  (/) ) )
98elrab 3257 . . 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 
(/) ) ) )
113, 4elmapd 7452 . . 3  |-  ( ph  ->  ( F  e.  ( A  ^m  B )  <-> 
F : B --> A ) )
1211anbi1d 704 . 2  |-  ( ph  ->  ( ( F  e.  ( A  ^m  B
)  /\  F finSupp  (/) )  <->  ( F : B --> A  /\  F finSupp  (/) ) ) )
1310, 12bitrd 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 1395    e. wcel 1819   {crab 2811   (/)c0 3793   class class class wbr 4456   Oncon0 4887   dom cdm 5008   -->wf 5590  (class class class)co 6296    ^m cmap 7438   finSupp cfsupp 7847   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-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-ov 6299  df-oprab 6300  df-mpt2 6301  df-recs 7060  df-rdg 7094  df-seqom 7131  df-map 7440  df-cnf 8096
This theorem is referenced by:  cantnfcl  8103  cantnfle  8107  cantnflt  8108  cantnff  8110  cantnf0  8111  cantnfrescl  8112  cantnfp1lem1  8114  cantnfp1lem2  8115  cantnfp1lem3  8116  cantnfp1  8117  oemapvali  8120  cantnflem1a  8121  cantnflem1b  8122  cantnflem1c  8123  cantnflem1d  8124  cantnflem1  8125  cantnflem3  8127  cantnf  8129  cnfcomlem  8160  cnfcom  8161  cnfcom2lem  8162  cnfcom3lem  8164  cnfcom3  8165
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