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Theorem cantnfrescl 7378
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 451 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  ( D  \  B ) )  ->  (/)  e.  A
)
41, 3eqeltrd 2357 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( D  \  B ) )  ->  X  e.  A )
54ralrimiva 2626 . . . . . . 7  |-  ( ph  ->  A. n  e.  ( D  \  B ) X  e.  A )
65biantrud 493 . . . . . 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 3356 . . . . . 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 254 . . . . 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 3534 . . . . . . 7  |-  ( B 
C_  D  <->  ( B  u.  ( D  \  B
) )  =  D )
119, 10sylib 188 . . . . . 6  |-  ( ph  ->  ( B  u.  ( D  \  B ) )  =  D )
1211raleqdv 2742 . . . . 5  |-  ( ph  ->  ( A. n  e.  ( B  u.  ( D  \  B ) ) X  e.  A  <->  A. n  e.  D  X  e.  A ) )
138, 12bitrd 244 . . . 4  |-  ( ph  ->  ( A. n  e.  B  X  e.  A  <->  A. n  e.  D  X  e.  A ) )
14 eqid 2283 . . . . 5  |-  ( n  e.  B  |->  X )  =  ( n  e.  B  |->  X )
1514fmpt 5681 . . . 4  |-  ( A. n  e.  B  X  e.  A  <->  ( n  e.  B  |->  X ) : B --> A )
16 eqid 2283 . . . . 5  |-  ( n  e.  D  |->  X )  =  ( n  e.  D  |->  X )
1716fmpt 5681 . . . 4  |-  ( A. n  e.  D  X  e.  A  <->  ( n  e.  D  |->  X ) : D --> A )
1813, 15, 173bitr3g 278 . . 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 7377 . . . 4  |-  ( ph  ->  ( `' ( n  e.  B  |->  X )
" ( _V  \  1o ) )  =  ( `' ( n  e.  D  |->  X ) "
( _V  \  1o ) ) )
2423eleq1d 2349 . . 3  |-  ( ph  ->  ( ( `' ( n  e.  B  |->  X ) " ( _V 
\  1o ) )  e.  Fin  <->  ( `' ( n  e.  D  |->  X ) " ( _V  \  1o ) )  e.  Fin ) )
2518, 24anbi12d 691 . 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 7367 . 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 7367 . 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 276 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 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   _Vcvv 2788    \ cdif 3149    u. cun 3150    C_ wss 3152   (/)c0 3455    e. cmpt 4077   Oncon0 4392   `'ccnv 4688   dom cdm 4689   "cima 4692   -->wf 5251  (class class class)co 5858   1oc1o 6472   Fincfn 6863   CNF ccnf 7362
This theorem is referenced by:  cantnfres  7379
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-riota 6304  df-recs 6388  df-rdg 6423  df-seqom 6460  df-1o 6479  df-map 6774  df-oi 7225  df-cnf 7363
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