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Theorem cantnfreslem 7377
Description: The support of an extended function is the same as the original. (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  =  (/) )
Assertion
Ref Expression
cantnfreslem  |-  ( ph  ->  ( `' ( n  e.  B  |->  X )
" ( _V  \  1o ) )  =  ( `' ( n  e.  D  |->  X ) "
( _V  \  1o ) ) )
Distinct variable groups:    B, n    D, n    A, n    ph, n
Allowed substitution hints:    S( n)    X( n)

Proof of Theorem cantnfreslem
StepHypRef Expression
1 cantnfres.6 . . . . . . 7  |-  ( ph  ->  B  C_  D )
21sseld 3179 . . . . . 6  |-  ( ph  ->  ( n  e.  B  ->  n  e.  D ) )
32anim1d 547 . . . . 5  |-  ( ph  ->  ( ( n  e.  B  /\  X  e.  ( _V  \  1o ) )  ->  (
n  e.  D  /\  X  e.  ( _V  \  1o ) ) ) )
4 eldif 3162 . . . . . . . . . . . 12  |-  ( n  e.  ( D  \  B )  <->  ( n  e.  D  /\  -.  n  e.  B ) )
5 cantnfres.7 . . . . . . . . . . . 12  |-  ( (
ph  /\  n  e.  ( D  \  B ) )  ->  X  =  (/) )
64, 5sylan2br 462 . . . . . . . . . . 11  |-  ( (
ph  /\  ( n  e.  D  /\  -.  n  e.  B ) )  ->  X  =  (/) )
76expr 598 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  D )  ->  ( -.  n  e.  B  ->  X  =  (/) ) )
8 el1o 6498 . . . . . . . . . . 11  |-  ( X  e.  1o  <->  X  =  (/) )
9 elndif 3300 . . . . . . . . . . 11  |-  ( X  e.  1o  ->  -.  X  e.  ( _V  \  1o ) )
108, 9sylbir 204 . . . . . . . . . 10  |-  ( X  =  (/)  ->  -.  X  e.  ( _V  \  1o ) )
117, 10syl6 29 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  D )  ->  ( -.  n  e.  B  ->  -.  X  e.  ( _V  \  1o ) ) )
1211con4d 97 . . . . . . . 8  |-  ( (
ph  /\  n  e.  D )  ->  ( X  e.  ( _V  \  1o )  ->  n  e.  B ) )
1312impr 602 . . . . . . 7  |-  ( (
ph  /\  ( n  e.  D  /\  X  e.  ( _V  \  1o ) ) )  ->  n  e.  B )
14 simprr 733 . . . . . . 7  |-  ( (
ph  /\  ( n  e.  D  /\  X  e.  ( _V  \  1o ) ) )  ->  X  e.  ( _V  \  1o ) )
1513, 14jca 518 . . . . . 6  |-  ( (
ph  /\  ( n  e.  D  /\  X  e.  ( _V  \  1o ) ) )  -> 
( n  e.  B  /\  X  e.  ( _V  \  1o ) ) )
1615ex 423 . . . . 5  |-  ( ph  ->  ( ( n  e.  D  /\  X  e.  ( _V  \  1o ) )  ->  (
n  e.  B  /\  X  e.  ( _V  \  1o ) ) ) )
173, 16impbid 183 . . . 4  |-  ( ph  ->  ( ( n  e.  B  /\  X  e.  ( _V  \  1o ) )  <->  ( n  e.  D  /\  X  e.  ( _V  \  1o ) ) ) )
1817abbidv 2397 . . 3  |-  ( ph  ->  { n  |  ( n  e.  B  /\  X  e.  ( _V  \  1o ) ) }  =  { n  |  ( n  e.  D  /\  X  e.  ( _V  \  1o ) ) } )
19 df-rab 2552 . . 3  |-  { n  e.  B  |  X  e.  ( _V  \  1o ) }  =  {
n  |  ( n  e.  B  /\  X  e.  ( _V  \  1o ) ) }
20 df-rab 2552 . . 3  |-  { n  e.  D  |  X  e.  ( _V  \  1o ) }  =  {
n  |  ( n  e.  D  /\  X  e.  ( _V  \  1o ) ) }
2118, 19, 203eqtr4g 2340 . 2  |-  ( ph  ->  { n  e.  B  |  X  e.  ( _V  \  1o ) }  =  { n  e.  D  |  X  e.  ( _V  \  1o ) } )
22 eqid 2283 . . 3  |-  ( n  e.  B  |->  X )  =  ( n  e.  B  |->  X )
2322mptpreima 5166 . 2  |-  ( `' ( n  e.  B  |->  X ) " ( _V  \  1o ) )  =  { n  e.  B  |  X  e.  ( _V  \  1o ) }
24 eqid 2283 . . 3  |-  ( n  e.  D  |->  X )  =  ( n  e.  D  |->  X )
2524mptpreima 5166 . 2  |-  ( `' ( n  e.  D  |->  X ) " ( _V  \  1o ) )  =  { n  e.  D  |  X  e.  ( _V  \  1o ) }
2621, 23, 253eqtr4g 2340 1  |-  ( ph  ->  ( `' ( n  e.  B  |->  X )
" ( _V  \  1o ) )  =  ( `' ( n  e.  D  |->  X ) "
( _V  \  1o ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   {cab 2269   {crab 2547   _Vcvv 2788    \ cdif 3149    C_ wss 3152   (/)c0 3455    e. cmpt 4077   Oncon0 4392   `'ccnv 4688   dom cdm 4689   "cima 4692  (class class class)co 5858   1oc1o 6472   CNF ccnf 7362
This theorem is referenced by:  cantnfrescl  7378  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-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  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-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-mpt 4079  df-suc 4398  df-xp 4695  df-rel 4696  df-cnv 4697  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-1o 6479
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