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Theorem cantnfs 7984
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 2454 . . . . . 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 7980 . . . . 5  |-  ( ph  ->  dom  ( A CNF  B
)  =  { g  e.  ( A  ^m  B )  |  g finSupp  (/)
} )
61, 5syl5eq 2507 . . . 4  |-  ( ph  ->  S  =  { g  e.  ( A  ^m  B )  |  g finSupp  (/)
} )
76eleq2d 2524 . . 3  |-  ( ph  ->  ( F  e.  S  <->  F  e.  { g  e.  ( A  ^m  B
)  |  g finSupp  (/) } ) )
8 breq1 4402 . . . 4  |-  ( g  =  F  ->  (
g finSupp  (/)  <->  F finSupp  (/) ) )
98elrab 3222 . . 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 7336 . . . 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 1370    e. wcel 1758   {crab 2802   (/)c0 3744   class class class wbr 4399   Oncon0 4826   dom cdm 4947   -->wf 5521  (class class class)co 6199    ^m cmap 7323   finSupp cfsupp 7730   CNF ccnf 7977
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 1955  ax-ext 2432  ax-rep 4510  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481
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 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-ral 2803  df-rex 2804  df-reu 2805  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-op 3991  df-uni 4199  df-iun 4280  df-br 4400  df-opab 4458  df-mpt 4459  df-id 4743  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-ov 6202  df-oprab 6203  df-mpt2 6204  df-recs 6941  df-rdg 6975  df-seqom 7012  df-map 7325  df-cnf 7978
This theorem is referenced by:  cantnfcl  7985  cantnfle  7989  cantnflt  7990  cantnff  7992  cantnf0  7993  cantnfrescl  7994  cantnfp1lem1  7996  cantnfp1lem2  7997  cantnfp1lem3  7998  cantnfp1  7999  oemapvali  8002  cantnflem1a  8003  cantnflem1b  8004  cantnflem1c  8005  cantnflem1d  8006  cantnflem1  8007  cantnflem3  8009  cantnf  8011  cnfcomlem  8042  cnfcom  8043  cnfcom2lem  8044  cnfcom3lem  8046  cnfcom3  8047
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