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Theorem nfvres 5321
Description: The value of a non-member of a restriction is the empty set. (Contributed by NM, 13-Nov-1995.)
Assertion
Ref Expression
nfvres  |-  ( -.  A  e.  B  -> 
( ( F  |`  B ) `  A
)  =  (/) )

Proof of Theorem nfvres
Dummy variables  x  y  z  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-fv 5010 . . . . . . . . . 10  |-  ( ( F  |`  B ) `  A )  =  ( iota x A ( F  |`  B )
x )
2 df-iota 4967 . . . . . . . . . 10  |-  ( iota
x A ( F  |`  B ) x )  =  U. { y  |  { x  |  A ( F  |`  B ) x }  =  { y } }
31, 2eqtri 2108 . . . . . . . . 9  |-  ( ( F  |`  B ) `  A )  =  U. { y  |  {
x  |  A ( F  |`  B )
x }  =  {
y } }
43eleq2i 2154 . . . . . . . 8  |-  ( z  e.  ( ( F  |`  B ) `  A
)  <->  z  e.  U. { y  |  {
x  |  A ( F  |`  B )
x }  =  {
y } } )
5 eluni 3651 . . . . . . . 8  |-  ( z  e.  U. { y  |  { x  |  A ( F  |`  B ) x }  =  { y } }  <->  E. w ( z  e.  w  /\  w  e. 
{ y  |  {
x  |  A ( F  |`  B )
x }  =  {
y } } ) )
64, 5bitri 182 . . . . . . 7  |-  ( z  e.  ( ( F  |`  B ) `  A
)  <->  E. w ( z  e.  w  /\  w  e.  { y  |  {
x  |  A ( F  |`  B )
x }  =  {
y } } ) )
7 exsimpr 1554 . . . . . . 7  |-  ( E. w ( z  e.  w  /\  w  e. 
{ y  |  {
x  |  A ( F  |`  B )
x }  =  {
y } } )  ->  E. w  w  e. 
{ y  |  {
x  |  A ( F  |`  B )
x }  =  {
y } } )
86, 7sylbi 119 . . . . . 6  |-  ( z  e.  ( ( F  |`  B ) `  A
)  ->  E. w  w  e.  { y  |  { x  |  A
( F  |`  B ) x }  =  {
y } } )
9 df-clab 2075 . . . . . . . 8  |-  ( w  e.  { y  |  { x  |  A
( F  |`  B ) x }  =  {
y } }  <->  [ w  /  y ] {
x  |  A ( F  |`  B )
x }  =  {
y } )
10 nfv 1466 . . . . . . . . 9  |-  F/ y { x  |  A
( F  |`  B ) x }  =  {
w }
11 sneq 3452 . . . . . . . . . 10  |-  ( y  =  w  ->  { y }  =  { w } )
1211eqeq2d 2099 . . . . . . . . 9  |-  ( y  =  w  ->  ( { x  |  A
( F  |`  B ) x }  =  {
y }  <->  { x  |  A ( F  |`  B ) x }  =  { w } ) )
1310, 12sbie 1721 . . . . . . . 8  |-  ( [ w  /  y ] { x  |  A
( F  |`  B ) x }  =  {
y }  <->  { x  |  A ( F  |`  B ) x }  =  { w } )
149, 13bitri 182 . . . . . . 7  |-  ( w  e.  { y  |  { x  |  A
( F  |`  B ) x }  =  {
y } }  <->  { x  |  A ( F  |`  B ) x }  =  { w } )
1514exbii 1541 . . . . . 6  |-  ( E. w  w  e.  {
y  |  { x  |  A ( F  |`  B ) x }  =  { y } }  <->  E. w { x  |  A ( F  |`  B ) x }  =  { w } )
168, 15sylib 120 . . . . 5  |-  ( z  e.  ( ( F  |`  B ) `  A
)  ->  E. w { x  |  A
( F  |`  B ) x }  =  {
w } )
17 euabsn2 3506 . . . . 5  |-  ( E! x  A ( F  |`  B ) x  <->  E. w { x  |  A
( F  |`  B ) x }  =  {
w } )
1816, 17sylibr 132 . . . 4  |-  ( z  e.  ( ( F  |`  B ) `  A
)  ->  E! x  A ( F  |`  B ) x )
19 euex 1978 . . . 4  |-  ( E! x  A ( F  |`  B ) x  ->  E. x  A ( F  |`  B ) x )
20 df-br 3838 . . . . . . . 8  |-  ( A ( F  |`  B ) x  <->  <. A ,  x >.  e.  ( F  |`  B ) )
21 df-res 4440 . . . . . . . . 9  |-  ( F  |`  B )  =  ( F  i^i  ( B  X.  _V ) )
2221eleq2i 2154 . . . . . . . 8  |-  ( <. A ,  x >.  e.  ( F  |`  B )  <->  <. A ,  x >.  e.  ( F  i^i  ( B  X.  _V ) ) )
2320, 22bitri 182 . . . . . . 7  |-  ( A ( F  |`  B ) x  <->  <. A ,  x >.  e.  ( F  i^i  ( B  X.  _V )
) )
24 elin 3181 . . . . . . . 8  |-  ( <. A ,  x >.  e.  ( F  i^i  ( B  X.  _V ) )  <-> 
( <. A ,  x >.  e.  F  /\  <. A ,  x >.  e.  ( B  X.  _V )
) )
2524simprbi 269 . . . . . . 7  |-  ( <. A ,  x >.  e.  ( F  i^i  ( B  X.  _V ) )  ->  <. A ,  x >.  e.  ( B  X.  _V ) )
2623, 25sylbi 119 . . . . . 6  |-  ( A ( F  |`  B ) x  ->  <. A ,  x >.  e.  ( B  X.  _V ) )
27 opelxp1 4460 . . . . . 6  |-  ( <. A ,  x >.  e.  ( B  X.  _V )  ->  A  e.  B
)
2826, 27syl 14 . . . . 5  |-  ( A ( F  |`  B ) x  ->  A  e.  B )
2928exlimiv 1534 . . . 4  |-  ( E. x  A ( F  |`  B ) x  ->  A  e.  B )
3018, 19, 293syl 17 . . 3  |-  ( z  e.  ( ( F  |`  B ) `  A
)  ->  A  e.  B )
3130con3i 597 . 2  |-  ( -.  A  e.  B  ->  -.  z  e.  (
( F  |`  B ) `
 A ) )
3231eq0rdv 3324 1  |-  ( -.  A  e.  B  -> 
( ( F  |`  B ) `  A
)  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    = wceq 1289   E.wex 1426    e. wcel 1438   [wsb 1692   E!weu 1948   {cab 2074   _Vcvv 2619    i^i cin 2996   (/)c0 3284   {csn 3441   <.cop 3444   U.cuni 3648   class class class wbr 3837    X. cxp 4426    |` cres 4430   iotacio 4965   ` cfv 5002
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-br 3838  df-opab 3892  df-xp 4434  df-res 4440  df-iota 4967  df-fv 5010
This theorem is referenced by: (None)
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