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Theorem cantnfreslem 7393
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 3192 . . . . . 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 3175 . . . . . . . . . . . 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 6514 . . . . . . . . . . 11  |-  ( X  e.  1o  <->  X  =  (/) )
9 elndif 3313 . . . . . . . . . . 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 2410 . . 3  |-  ( ph  ->  { n  |  ( n  e.  B  /\  X  e.  ( _V  \  1o ) ) }  =  { n  |  ( n  e.  D  /\  X  e.  ( _V  \  1o ) ) } )
19 df-rab 2565 . . 3  |-  { n  e.  B  |  X  e.  ( _V  \  1o ) }  =  {
n  |  ( n  e.  B  /\  X  e.  ( _V  \  1o ) ) }
20 df-rab 2565 . . 3  |-  { n  e.  D  |  X  e.  ( _V  \  1o ) }  =  {
n  |  ( n  e.  D  /\  X  e.  ( _V  \  1o ) ) }
2118, 19, 203eqtr4g 2353 . 2  |-  ( ph  ->  { n  e.  B  |  X  e.  ( _V  \  1o ) }  =  { n  e.  D  |  X  e.  ( _V  \  1o ) } )
22 eqid 2296 . . 3  |-  ( n  e.  B  |->  X )  =  ( n  e.  B  |->  X )
2322mptpreima 5182 . 2  |-  ( `' ( n  e.  B  |->  X ) " ( _V  \  1o ) )  =  { n  e.  B  |  X  e.  ( _V  \  1o ) }
24 eqid 2296 . . 3  |-  ( n  e.  D  |->  X )  =  ( n  e.  D  |->  X )
2524mptpreima 5182 . 2  |-  ( `' ( n  e.  D  |->  X ) " ( _V  \  1o ) )  =  { n  e.  D  |  X  e.  ( _V  \  1o ) }
2621, 23, 253eqtr4g 2353 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 1632    e. wcel 1696   {cab 2282   {crab 2560   _Vcvv 2801    \ cdif 3162    C_ wss 3165   (/)c0 3468    e. cmpt 4093   Oncon0 4408   `'ccnv 4704   dom cdm 4705   "cima 4708  (class class class)co 5874   1oc1o 6488   CNF ccnf 7378
This theorem is referenced by:  cantnfrescl  7394  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-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  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-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-mpt 4095  df-suc 4414  df-xp 4711  df-rel 4712  df-cnv 4713  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-1o 6495
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