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Theorem cantnfrescl 7634
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.1  |-  S  =  dom  ( A CNF  B
)
cantnfs.2  |-  ( ph  ->  A  e.  On )
cantnfs.3  |-  ( ph  ->  B  e.  On )
cantnfres.5  |-  ( ph  ->  D  e.  On )
cantnfres.6  |-  ( ph  ->  B  C_  D )
cantnfres.7  |-  ( (
ph  /\  n  e.  ( D  \  B ) )  ->  X  =  (/) )
cantnfres.8  |-  ( ph  -> 
(/)  e.  A )
cantnfres.9  |-  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 cantnfres.7 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  ( D  \  B ) )  ->  X  =  (/) )
2 cantnfres.8 . . . . . . . . . 10  |-  ( ph  -> 
(/)  e.  A )
32adantr 453 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  ( D  \  B ) )  ->  (/)  e.  A
)
41, 3eqeltrd 2512 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( D  \  B ) )  ->  X  e.  A )
54ralrimiva 2791 . . . . . . 7  |-  ( ph  ->  A. n  e.  ( D  \  B ) X  e.  A )
65biantrud 495 . . . . . 6  |-  ( ph  ->  ( A. n  e.  B  X  e.  A  <->  ( A. n  e.  B  X  e.  A  /\  A. n  e.  ( D 
\  B ) X  e.  A ) ) )
7 ralunb 3530 . . . . . 6  |-  ( A. n  e.  ( B  u.  ( D  \  B
) ) X  e.  A  <->  ( A. n  e.  B  X  e.  A  /\  A. n  e.  ( D  \  B
) X  e.  A
) )
86, 7syl6bbr 256 . . . . 5  |-  ( ph  ->  ( A. n  e.  B  X  e.  A  <->  A. n  e.  ( B  u.  ( D  \  B ) ) X  e.  A ) )
9 cantnfres.6 . . . . . . 7  |-  ( ph  ->  B  C_  D )
10 undif 3710 . . . . . . 7  |-  ( B 
C_  D  <->  ( B  u.  ( D  \  B
) )  =  D )
119, 10sylib 190 . . . . . 6  |-  ( ph  ->  ( B  u.  ( D  \  B ) )  =  D )
1211raleqdv 2912 . . . . 5  |-  ( ph  ->  ( A. n  e.  ( B  u.  ( D  \  B ) ) X  e.  A  <->  A. n  e.  D  X  e.  A ) )
138, 12bitrd 246 . . . 4  |-  ( ph  ->  ( A. n  e.  B  X  e.  A  <->  A. n  e.  D  X  e.  A ) )
14 eqid 2438 . . . . 5  |-  ( n  e.  B  |->  X )  =  ( n  e.  B  |->  X )
1514fmpt 5892 . . . 4  |-  ( A. n  e.  B  X  e.  A  <->  ( n  e.  B  |->  X ) : B --> A )
16 eqid 2438 . . . . 5  |-  ( n  e.  D  |->  X )  =  ( n  e.  D  |->  X )
1716fmpt 5892 . . . 4  |-  ( A. n  e.  D  X  e.  A  <->  ( n  e.  D  |->  X ) : D --> A )
1813, 15, 173bitr3g 280 . . 3  |-  ( ph  ->  ( ( n  e.  B  |->  X ) : B --> A  <->  ( n  e.  D  |->  X ) : D --> A ) )
19 cantnfs.1 . . . . 5  |-  S  =  dom  ( A CNF  B
)
20 cantnfs.2 . . . . 5  |-  ( ph  ->  A  e.  On )
21 cantnfs.3 . . . . 5  |-  ( ph  ->  B  e.  On )
22 cantnfres.5 . . . . 5  |-  ( ph  ->  D  e.  On )
2319, 20, 21, 22, 9, 1cantnfreslem 7633 . . . 4  |-  ( ph  ->  ( `' ( n  e.  B  |->  X )
" ( _V  \  1o ) )  =  ( `' ( n  e.  D  |->  X ) "
( _V  \  1o ) ) )
2423eleq1d 2504 . . 3  |-  ( ph  ->  ( ( `' ( n  e.  B  |->  X ) " ( _V 
\  1o ) )  e.  Fin  <->  ( `' ( n  e.  D  |->  X ) " ( _V  \  1o ) )  e.  Fin ) )
2518, 24anbi12d 693 . 2  |-  ( ph  ->  ( ( ( n  e.  B  |->  X ) : B --> A  /\  ( `' ( n  e.  B  |->  X ) "
( _V  \  1o ) )  e.  Fin ) 
<->  ( ( n  e.  D  |->  X ) : D --> A  /\  ( `' ( n  e.  D  |->  X ) "
( _V  \  1o ) )  e.  Fin ) ) )
2619, 20, 21cantnfs 7623 . 2  |-  ( ph  ->  ( ( n  e.  B  |->  X )  e.  S  <->  ( ( n  e.  B  |->  X ) : B --> A  /\  ( `' ( n  e.  B  |->  X ) "
( _V  \  1o ) )  e.  Fin ) ) )
27 cantnfres.9 . . 3  |-  T  =  dom  ( A CNF  D
)
2827, 20, 22cantnfs 7623 . 2  |-  ( ph  ->  ( ( n  e.  D  |->  X )  e.  T  <->  ( ( n  e.  D  |->  X ) : D --> A  /\  ( `' ( n  e.  D  |->  X ) "
( _V  \  1o ) )  e.  Fin ) ) )
2925, 26, 283bitr4d 278 1  |-  ( ph  ->  ( ( n  e.  B  |->  X )  e.  S  <->  ( n  e.  D  |->  X )  e.  T ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2707   _Vcvv 2958    \ cdif 3319    u. cun 3320    C_ wss 3322   (/)c0 3630    e. cmpt 4268   Oncon0 4583   `'ccnv 4879   dom cdm 4880   "cima 4883   -->wf 5452  (class class class)co 6083   1oc1o 6719   Fincfn 7111   CNF ccnf 7618
This theorem is referenced by:  cantnfres  7635
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-se 4544  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-isom 5465  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-riota 6551  df-recs 6635  df-rdg 6670  df-seqom 6707  df-1o 6726  df-map 7022  df-oi 7481  df-cnf 7619
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