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Theorem cantnfrescl 8047
Description: A function is finitely supported from  B to  A iff the extended function is finitely supported from  D to  A. (Contributed by Mario Carneiro, 25-May-2015.)
Hypotheses
Ref Expression
cantnfs.s  |-  S  =  dom  ( A CNF  B
)
cantnfs.a  |-  ( ph  ->  A  e.  On )
cantnfs.b  |-  ( ph  ->  B  e.  On )
cantnfrescl.d  |-  ( ph  ->  D  e.  On )
cantnfrescl.b  |-  ( ph  ->  B  C_  D )
cantnfrescl.x  |-  ( (
ph  /\  n  e.  ( D  \  B ) )  ->  X  =  (/) )
cantnfrescl.a  |-  ( ph  -> 
(/)  e.  A )
cantnfrescl.t  |-  T  =  dom  ( A CNF  D
)
Assertion
Ref Expression
cantnfrescl  |-  ( ph  ->  ( ( n  e.  B  |->  X )  e.  S  <->  ( n  e.  D  |->  X )  e.  T ) )
Distinct variable groups:    B, n    D, n    A, n    ph, n
Allowed substitution hints:    S( n)    T( n)    X( n)

Proof of Theorem cantnfrescl
StepHypRef Expression
1 cantnfrescl.b . . . . 5  |-  ( ph  ->  B  C_  D )
2 cantnfrescl.x . . . . . . 7  |-  ( (
ph  /\  n  e.  ( D  \  B ) )  ->  X  =  (/) )
3 cantnfrescl.a . . . . . . . 8  |-  ( ph  -> 
(/)  e.  A )
43adantr 463 . . . . . . 7  |-  ( (
ph  /\  n  e.  ( D  \  B ) )  ->  (/)  e.  A
)
52, 4eqeltrd 2490 . . . . . 6  |-  ( (
ph  /\  n  e.  ( D  \  B ) )  ->  X  e.  A )
65ralrimiva 2817 . . . . 5  |-  ( ph  ->  A. n  e.  ( D  \  B ) X  e.  A )
71, 6raldifeq 3860 . . . 4  |-  ( ph  ->  ( A. n  e.  B  X  e.  A  <->  A. n  e.  D  X  e.  A ) )
8 eqid 2402 . . . . 5  |-  ( n  e.  B  |->  X )  =  ( n  e.  B  |->  X )
98fmpt 5986 . . . 4  |-  ( A. n  e.  B  X  e.  A  <->  ( n  e.  B  |->  X ) : B --> A )
10 eqid 2402 . . . . 5  |-  ( n  e.  D  |->  X )  =  ( n  e.  D  |->  X )
1110fmpt 5986 . . . 4  |-  ( A. n  e.  D  X  e.  A  <->  ( n  e.  D  |->  X ) : D --> A )
127, 9, 113bitr3g 287 . . 3  |-  ( ph  ->  ( ( n  e.  B  |->  X ) : B --> A  <->  ( n  e.  D  |->  X ) : D --> A ) )
13 cantnfs.b . . . . . 6  |-  ( ph  ->  B  e.  On )
14 mptexg 6079 . . . . . 6  |-  ( B  e.  On  ->  (
n  e.  B  |->  X )  e.  _V )
1513, 14syl 17 . . . . 5  |-  ( ph  ->  ( n  e.  B  |->  X )  e.  _V )
16 funmpt 5561 . . . . . 6  |-  Fun  (
n  e.  B  |->  X )
1716a1i 11 . . . . 5  |-  ( ph  ->  Fun  ( n  e.  B  |->  X ) )
18 cantnfrescl.d . . . . . . 7  |-  ( ph  ->  D  e.  On )
19 mptexg 6079 . . . . . . 7  |-  ( D  e.  On  ->  (
n  e.  D  |->  X )  e.  _V )
2018, 19syl 17 . . . . . 6  |-  ( ph  ->  ( n  e.  D  |->  X )  e.  _V )
21 funmpt 5561 . . . . . 6  |-  Fun  (
n  e.  D  |->  X )
2220, 21jctir 536 . . . . 5  |-  ( ph  ->  ( ( n  e.  D  |->  X )  e. 
_V  /\  Fun  ( n  e.  D  |->  X ) ) )
2315, 17, 22jca31 532 . . . 4  |-  ( ph  ->  ( ( ( n  e.  B  |->  X )  e.  _V  /\  Fun  ( n  e.  B  |->  X ) )  /\  ( ( n  e.  D  |->  X )  e. 
_V  /\  Fun  ( n  e.  D  |->  X ) ) ) )
2418, 1, 2extmptsuppeq 6881 . . . 4  |-  ( ph  ->  ( ( n  e.  B  |->  X ) supp  (/) )  =  ( ( n  e.  D  |->  X ) supp  (/) ) )
25 suppeqfsuppbi 7797 . . . 4  |-  ( ( ( ( n  e.  B  |->  X )  e. 
_V  /\  Fun  ( n  e.  B  |->  X ) )  /\  ( ( n  e.  D  |->  X )  e.  _V  /\  Fun  ( n  e.  D  |->  X ) ) )  ->  ( ( ( n  e.  B  |->  X ) supp  (/) )  =  ( ( n  e.  D  |->  X ) supp  (/) )  -> 
( ( n  e.  B  |->  X ) finSupp  (/)  <->  ( n  e.  D  |->  X ) finSupp  (/) ) ) )
2623, 24, 25sylc 59 . . 3  |-  ( ph  ->  ( ( n  e.  B  |->  X ) finSupp  (/)  <->  ( n  e.  D  |->  X ) finSupp  (/) ) )
2712, 26anbi12d 709 . 2  |-  ( ph  ->  ( ( ( n  e.  B  |->  X ) : B --> A  /\  ( n  e.  B  |->  X ) finSupp  (/) )  <->  ( (
n  e.  D  |->  X ) : D --> A  /\  ( n  e.  D  |->  X ) finSupp  (/) ) ) )
28 cantnfs.s . . 3  |-  S  =  dom  ( A CNF  B
)
29 cantnfs.a . . 3  |-  ( ph  ->  A  e.  On )
3028, 29, 13cantnfs 8037 . 2  |-  ( ph  ->  ( ( n  e.  B  |->  X )  e.  S  <->  ( ( n  e.  B  |->  X ) : B --> A  /\  ( n  e.  B  |->  X ) finSupp  (/) ) ) )
31 cantnfrescl.t . . 3  |-  T  =  dom  ( A CNF  D
)
3231, 29, 18cantnfs 8037 . 2  |-  ( ph  ->  ( ( n  e.  D  |->  X )  e.  T  <->  ( ( n  e.  D  |->  X ) : D --> A  /\  ( n  e.  D  |->  X ) finSupp  (/) ) ) )
3327, 30, 323bitr4d 285 1  |-  ( ph  ->  ( ( n  e.  B  |->  X )  e.  S  <->  ( n  e.  D  |->  X )  e.  T ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1405    e. wcel 1842   A.wral 2753   _Vcvv 3058    \ cdif 3410    C_ wss 3413   (/)c0 3737   class class class wbr 4394    |-> cmpt 4452   Oncon0 4821   dom cdm 4942   Fun wfun 5519   -->wf 5521  (class class class)co 6234   supp csupp 6856   finSupp cfsupp 7783   CNF ccnf 8030
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6530
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-fal 1411  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-reu 2760  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-rn 4953  df-res 4954  df-ima 4955  df-iota 5489  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-ov 6237  df-oprab 6238  df-mpt2 6239  df-supp 6857  df-recs 6999  df-rdg 7033  df-seqom 7070  df-map 7379  df-fsupp 7784  df-cnf 8031
This theorem is referenced by:  cantnfres  8048
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