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Theorem cantnfrescl 8181
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 467 . . . . . . 7  |-  ( (
ph  /\  n  e.  ( D  \  B ) )  ->  (/)  e.  A
)
52, 4eqeltrd 2529 . . . . . 6  |-  ( (
ph  /\  n  e.  ( D  \  B ) )  ->  X  e.  A )
65ralrimiva 2802 . . . . 5  |-  ( ph  ->  A. n  e.  ( D  \  B ) X  e.  A )
71, 6raldifeq 3857 . . . 4  |-  ( ph  ->  ( A. n  e.  B  X  e.  A  <->  A. n  e.  D  X  e.  A ) )
8 eqid 2451 . . . . 5  |-  ( n  e.  B  |->  X )  =  ( n  e.  B  |->  X )
98fmpt 6043 . . . 4  |-  ( A. n  e.  B  X  e.  A  <->  ( n  e.  B  |->  X ) : B --> A )
10 eqid 2451 . . . . 5  |-  ( n  e.  D  |->  X )  =  ( n  e.  D  |->  X )
1110fmpt 6043 . . . 4  |-  ( A. n  e.  D  X  e.  A  <->  ( n  e.  D  |->  X ) : D --> A )
127, 9, 113bitr3g 291 . . 3  |-  ( ph  ->  ( ( n  e.  B  |->  X ) : B --> A  <->  ( n  e.  D  |->  X ) : D --> A ) )
13 cantnfs.b . . . . . 6  |-  ( ph  ->  B  e.  On )
14 mptexg 6135 . . . . . 6  |-  ( B  e.  On  ->  (
n  e.  B  |->  X )  e.  _V )
1513, 14syl 17 . . . . 5  |-  ( ph  ->  ( n  e.  B  |->  X )  e.  _V )
16 funmpt 5618 . . . . . 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 6135 . . . . . . 7  |-  ( D  e.  On  ->  (
n  e.  D  |->  X )  e.  _V )
2018, 19syl 17 . . . . . 6  |-  ( ph  ->  ( n  e.  D  |->  X )  e.  _V )
21 funmpt 5618 . . . . . 6  |-  Fun  (
n  e.  D  |->  X )
2220, 21jctir 541 . . . . 5  |-  ( ph  ->  ( ( n  e.  D  |->  X )  e. 
_V  /\  Fun  ( n  e.  D  |->  X ) ) )
2315, 17, 22jca31 537 . . . 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 6939 . . . 4  |-  ( ph  ->  ( ( n  e.  B  |->  X ) supp  (/) )  =  ( ( n  e.  D  |->  X ) supp  (/) ) )
25 suppeqfsuppbi 7897 . . . 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 62 . . 3  |-  ( ph  ->  ( ( n  e.  B  |->  X ) finSupp  (/)  <->  ( n  e.  D  |->  X ) finSupp  (/) ) )
2712, 26anbi12d 717 . 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 8171 . 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 8171 . 2  |-  ( ph  ->  ( ( n  e.  D  |->  X )  e.  T  <->  ( ( n  e.  D  |->  X ) : D --> A  /\  ( n  e.  D  |->  X ) finSupp  (/) ) ) )
3327, 30, 323bitr4d 289 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 188    /\ wa 371    = wceq 1444    e. wcel 1887   A.wral 2737   _Vcvv 3045    \ cdif 3401    C_ wss 3404   (/)c0 3731   class class class wbr 4402    |-> cmpt 4461   dom cdm 4834   Oncon0 5423   Fun wfun 5576   -->wf 5578  (class class class)co 6290   supp csupp 6914   finSupp cfsupp 7883   CNF ccnf 8166
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-fal 1450  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-reu 2744  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-supp 6915  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-seqom 7165  df-map 7474  df-fsupp 7884  df-cnf 8167
This theorem is referenced by:  cantnfres  8182
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