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Theorem cantnfreslem 7631
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 3347 . . . . . 6  |-  ( ph  ->  ( n  e.  B  ->  n  e.  D ) )
32anim1d 548 . . . . 5  |-  ( ph  ->  ( ( n  e.  B  /\  X  e.  ( _V  \  1o ) )  ->  (
n  e.  D  /\  X  e.  ( _V  \  1o ) ) ) )
4 eldif 3330 . . . . . . . . . . . 12  |-  ( n  e.  ( D  \  B )  <->  ( n  e.  D  /\  -.  n  e.  B ) )
5 cantnfres.7 . . . . . . . . . . . 12  |-  ( (
ph  /\  n  e.  ( D  \  B ) )  ->  X  =  (/) )
64, 5sylan2br 463 . . . . . . . . . . 11  |-  ( (
ph  /\  ( n  e.  D  /\  -.  n  e.  B ) )  ->  X  =  (/) )
76expr 599 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  D )  ->  ( -.  n  e.  B  ->  X  =  (/) ) )
8 el1o 6743 . . . . . . . . . . 11  |-  ( X  e.  1o  <->  X  =  (/) )
9 elndif 3471 . . . . . . . . . . 11  |-  ( X  e.  1o  ->  -.  X  e.  ( _V  \  1o ) )
108, 9sylbir 205 . . . . . . . . . 10  |-  ( X  =  (/)  ->  -.  X  e.  ( _V  \  1o ) )
117, 10syl6 31 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  D )  ->  ( -.  n  e.  B  ->  -.  X  e.  ( _V  \  1o ) ) )
1211con4d 99 . . . . . . . 8  |-  ( (
ph  /\  n  e.  D )  ->  ( X  e.  ( _V  \  1o )  ->  n  e.  B ) )
1312impr 603 . . . . . . 7  |-  ( (
ph  /\  ( n  e.  D  /\  X  e.  ( _V  \  1o ) ) )  ->  n  e.  B )
14 simprr 734 . . . . . . 7  |-  ( (
ph  /\  ( n  e.  D  /\  X  e.  ( _V  \  1o ) ) )  ->  X  e.  ( _V  \  1o ) )
1513, 14jca 519 . . . . . 6  |-  ( (
ph  /\  ( n  e.  D  /\  X  e.  ( _V  \  1o ) ) )  -> 
( n  e.  B  /\  X  e.  ( _V  \  1o ) ) )
1615ex 424 . . . . 5  |-  ( ph  ->  ( ( n  e.  D  /\  X  e.  ( _V  \  1o ) )  ->  (
n  e.  B  /\  X  e.  ( _V  \  1o ) ) ) )
173, 16impbid 184 . . . 4  |-  ( ph  ->  ( ( n  e.  B  /\  X  e.  ( _V  \  1o ) )  <->  ( n  e.  D  /\  X  e.  ( _V  \  1o ) ) ) )
1817abbidv 2550 . . 3  |-  ( ph  ->  { n  |  ( n  e.  B  /\  X  e.  ( _V  \  1o ) ) }  =  { n  |  ( n  e.  D  /\  X  e.  ( _V  \  1o ) ) } )
19 df-rab 2714 . . 3  |-  { n  e.  B  |  X  e.  ( _V  \  1o ) }  =  {
n  |  ( n  e.  B  /\  X  e.  ( _V  \  1o ) ) }
20 df-rab 2714 . . 3  |-  { n  e.  D  |  X  e.  ( _V  \  1o ) }  =  {
n  |  ( n  e.  D  /\  X  e.  ( _V  \  1o ) ) }
2118, 19, 203eqtr4g 2493 . 2  |-  ( ph  ->  { n  e.  B  |  X  e.  ( _V  \  1o ) }  =  { n  e.  D  |  X  e.  ( _V  \  1o ) } )
22 eqid 2436 . . 3  |-  ( n  e.  B  |->  X )  =  ( n  e.  B  |->  X )
2322mptpreima 5363 . 2  |-  ( `' ( n  e.  B  |->  X ) " ( _V  \  1o ) )  =  { n  e.  B  |  X  e.  ( _V  \  1o ) }
24 eqid 2436 . . 3  |-  ( n  e.  D  |->  X )  =  ( n  e.  D  |->  X )
2524mptpreima 5363 . 2  |-  ( `' ( n  e.  D  |->  X ) " ( _V  \  1o ) )  =  { n  e.  D  |  X  e.  ( _V  \  1o ) }
2621, 23, 253eqtr4g 2493 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 359    = wceq 1652    e. wcel 1725   {cab 2422   {crab 2709   _Vcvv 2956    \ cdif 3317    C_ wss 3320   (/)c0 3628    e. cmpt 4266   Oncon0 4581   `'ccnv 4877   dom cdm 4878   "cima 4881  (class class class)co 6081   1oc1o 6717   CNF ccnf 7616
This theorem is referenced by:  cantnfrescl  7632  cantnfres  7633
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-br 4213  df-opab 4267  df-mpt 4268  df-suc 4587  df-xp 4884  df-rel 4885  df-cnv 4886  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-1o 6724
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