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Theorem cantnfrescl 7394
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 2370 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( D  \  B ) )  ->  X  e.  A )
54ralrimiva 2639 . . . . . . 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 3369 . . . . . 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 3547 . . . . . . 7  |-  ( B 
C_  D  <->  ( B  u.  ( D  \  B
) )  =  D )
119, 10sylib 188 . . . . . 6  |-  ( ph  ->  ( B  u.  ( D  \  B ) )  =  D )
1211raleqdv 2755 . . . . 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 2296 . . . . 5  |-  ( n  e.  B  |->  X )  =  ( n  e.  B  |->  X )
1514fmpt 5697 . . . 4  |-  ( A. n  e.  B  X  e.  A  <->  ( n  e.  B  |->  X ) : B --> A )
16 eqid 2296 . . . . 5  |-  ( n  e.  D  |->  X )  =  ( n  e.  D  |->  X )
1716fmpt 5697 . . . 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 7393 . . . 4  |-  ( ph  ->  ( `' ( n  e.  B  |->  X )
" ( _V  \  1o ) )  =  ( `' ( n  e.  D  |->  X ) "
( _V  \  1o ) ) )
2423eleq1d 2362 . . 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 7383 . 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 7383 . 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 1632    e. wcel 1696   A.wral 2556   _Vcvv 2801    \ cdif 3162    u. cun 3163    C_ wss 3165   (/)c0 3468    e. cmpt 4093   Oncon0 4408   `'ccnv 4704   dom cdm 4705   "cima 4708   -->wf 5267  (class class class)co 5874   1oc1o 6488   Fincfn 6879   CNF ccnf 7378
This theorem is referenced by:  cantnfres  7395
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-riota 6320  df-recs 6404  df-rdg 6439  df-seqom 6476  df-1o 6495  df-map 6790  df-oi 7241  df-cnf 7379
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