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Theorem nfvres 5548
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 5224 . . . . . . . . . 10  |-  ( ( F  |`  B ) `  A )  =  ( iota x A ( F  |`  B )
x )
2 df-iota 5178 . . . . . . . . . 10  |-  ( iota
x A ( F  |`  B ) x )  =  U. { y  |  { x  |  A ( F  |`  B ) x }  =  { y } }
31, 2eqtri 2198 . . . . . . . . 9  |-  ( ( F  |`  B ) `  A )  =  U. { y  |  {
x  |  A ( F  |`  B )
x }  =  {
y } }
43eleq2i 2244 . . . . . . . 8  |-  ( z  e.  ( ( F  |`  B ) `  A
)  <->  z  e.  U. { y  |  {
x  |  A ( F  |`  B )
x }  =  {
y } } )
5 eluni 3812 . . . . . . . 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 184 . . . . . . 7  |-  ( z  e.  ( ( F  |`  B ) `  A
)  <->  E. w ( z  e.  w  /\  w  e.  { y  |  {
x  |  A ( F  |`  B )
x }  =  {
y } } ) )
7 exsimpr 1618 . . . . . . 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 121 . . . . . 6  |-  ( z  e.  ( ( F  |`  B ) `  A
)  ->  E. w  w  e.  { y  |  { x  |  A
( F  |`  B ) x }  =  {
y } } )
9 df-clab 2164 . . . . . . . 8  |-  ( w  e.  { y  |  { x  |  A
( F  |`  B ) x }  =  {
y } }  <->  [ w  /  y ] {
x  |  A ( F  |`  B )
x }  =  {
y } )
10 nfv 1528 . . . . . . . . 9  |-  F/ y { x  |  A
( F  |`  B ) x }  =  {
w }
11 sneq 3603 . . . . . . . . . 10  |-  ( y  =  w  ->  { y }  =  { w } )
1211eqeq2d 2189 . . . . . . . . 9  |-  ( y  =  w  ->  ( { x  |  A
( F  |`  B ) x }  =  {
y }  <->  { x  |  A ( F  |`  B ) x }  =  { w } ) )
1310, 12sbie 1791 . . . . . . . 8  |-  ( [ w  /  y ] { x  |  A
( F  |`  B ) x }  =  {
y }  <->  { x  |  A ( F  |`  B ) x }  =  { w } )
149, 13bitri 184 . . . . . . 7  |-  ( w  e.  { y  |  { x  |  A
( F  |`  B ) x }  =  {
y } }  <->  { x  |  A ( F  |`  B ) x }  =  { w } )
1514exbii 1605 . . . . . 6  |-  ( E. w  w  e.  {
y  |  { x  |  A ( F  |`  B ) x }  =  { y } }  <->  E. w { x  |  A ( F  |`  B ) x }  =  { w } )
168, 15sylib 122 . . . . 5  |-  ( z  e.  ( ( F  |`  B ) `  A
)  ->  E. w { x  |  A
( F  |`  B ) x }  =  {
w } )
17 euabsn2 3661 . . . . 5  |-  ( E! x  A ( F  |`  B ) x  <->  E. w { x  |  A
( F  |`  B ) x }  =  {
w } )
1816, 17sylibr 134 . . . 4  |-  ( z  e.  ( ( F  |`  B ) `  A
)  ->  E! x  A ( F  |`  B ) x )
19 euex 2056 . . . 4  |-  ( E! x  A ( F  |`  B ) x  ->  E. x  A ( F  |`  B ) x )
20 df-br 4004 . . . . . . . 8  |-  ( A ( F  |`  B ) x  <->  <. A ,  x >.  e.  ( F  |`  B ) )
21 df-res 4638 . . . . . . . . 9  |-  ( F  |`  B )  =  ( F  i^i  ( B  X.  _V ) )
2221eleq2i 2244 . . . . . . . 8  |-  ( <. A ,  x >.  e.  ( F  |`  B )  <->  <. A ,  x >.  e.  ( F  i^i  ( B  X.  _V ) ) )
2320, 22bitri 184 . . . . . . 7  |-  ( A ( F  |`  B ) x  <->  <. A ,  x >.  e.  ( F  i^i  ( B  X.  _V )
) )
24 elin 3318 . . . . . . . 8  |-  ( <. A ,  x >.  e.  ( F  i^i  ( B  X.  _V ) )  <-> 
( <. A ,  x >.  e.  F  /\  <. A ,  x >.  e.  ( B  X.  _V )
) )
2524simprbi 275 . . . . . . 7  |-  ( <. A ,  x >.  e.  ( F  i^i  ( B  X.  _V ) )  ->  <. A ,  x >.  e.  ( B  X.  _V ) )
2623, 25sylbi 121 . . . . . 6  |-  ( A ( F  |`  B ) x  ->  <. A ,  x >.  e.  ( B  X.  _V ) )
27 opelxp1 4660 . . . . . 6  |-  ( <. A ,  x >.  e.  ( B  X.  _V )  ->  A  e.  B
)
2826, 27syl 14 . . . . 5  |-  ( A ( F  |`  B ) x  ->  A  e.  B )
2928exlimiv 1598 . . . 4  |-  ( E. x  A ( F  |`  B ) x  ->  A  e.  B )
3018, 19, 293syl 17 . . 3  |-  ( z  e.  ( ( F  |`  B ) `  A
)  ->  A  e.  B )
3130con3i 632 . 2  |-  ( -.  A  e.  B  ->  -.  z  e.  (
( F  |`  B ) `
 A ) )
3231eq0rdv 3467 1  |-  ( -.  A  e.  B  -> 
( ( F  |`  B ) `  A
)  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 104    = wceq 1353   E.wex 1492   [wsb 1762   E!weu 2026    e. wcel 2148   {cab 2163   _Vcvv 2737    i^i cin 3128   (/)c0 3422   {csn 3592   <.cop 3595   U.cuni 3809   class class class wbr 4003    X. cxp 4624    |` cres 4628   iotacio 5176   ` cfv 5216
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4004  df-opab 4065  df-xp 4632  df-res 4638  df-iota 5178  df-fv 5224
This theorem is referenced by: (None)
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