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Theorem nfvres 5420
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 5099 . . . . . . . . . 10  |-  ( ( F  |`  B ) `  A )  =  ( iota x A ( F  |`  B )
x )
2 df-iota 5056 . . . . . . . . . 10  |-  ( iota
x A ( F  |`  B ) x )  =  U. { y  |  { x  |  A ( F  |`  B ) x }  =  { y } }
31, 2eqtri 2136 . . . . . . . . 9  |-  ( ( F  |`  B ) `  A )  =  U. { y  |  {
x  |  A ( F  |`  B )
x }  =  {
y } }
43eleq2i 2182 . . . . . . . 8  |-  ( z  e.  ( ( F  |`  B ) `  A
)  <->  z  e.  U. { y  |  {
x  |  A ( F  |`  B )
x }  =  {
y } } )
5 eluni 3707 . . . . . . . 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 183 . . . . . . 7  |-  ( z  e.  ( ( F  |`  B ) `  A
)  <->  E. w ( z  e.  w  /\  w  e.  { y  |  {
x  |  A ( F  |`  B )
x }  =  {
y } } ) )
7 exsimpr 1580 . . . . . . 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 120 . . . . . 6  |-  ( z  e.  ( ( F  |`  B ) `  A
)  ->  E. w  w  e.  { y  |  { x  |  A
( F  |`  B ) x }  =  {
y } } )
9 df-clab 2102 . . . . . . . 8  |-  ( w  e.  { y  |  { x  |  A
( F  |`  B ) x }  =  {
y } }  <->  [ w  /  y ] {
x  |  A ( F  |`  B )
x }  =  {
y } )
10 nfv 1491 . . . . . . . . 9  |-  F/ y { x  |  A
( F  |`  B ) x }  =  {
w }
11 sneq 3506 . . . . . . . . . 10  |-  ( y  =  w  ->  { y }  =  { w } )
1211eqeq2d 2127 . . . . . . . . 9  |-  ( y  =  w  ->  ( { x  |  A
( F  |`  B ) x }  =  {
y }  <->  { x  |  A ( F  |`  B ) x }  =  { w } ) )
1310, 12sbie 1747 . . . . . . . 8  |-  ( [ w  /  y ] { x  |  A
( F  |`  B ) x }  =  {
y }  <->  { x  |  A ( F  |`  B ) x }  =  { w } )
149, 13bitri 183 . . . . . . 7  |-  ( w  e.  { y  |  { x  |  A
( F  |`  B ) x }  =  {
y } }  <->  { x  |  A ( F  |`  B ) x }  =  { w } )
1514exbii 1567 . . . . . 6  |-  ( E. w  w  e.  {
y  |  { x  |  A ( F  |`  B ) x }  =  { y } }  <->  E. w { x  |  A ( F  |`  B ) x }  =  { w } )
168, 15sylib 121 . . . . 5  |-  ( z  e.  ( ( F  |`  B ) `  A
)  ->  E. w { x  |  A
( F  |`  B ) x }  =  {
w } )
17 euabsn2 3560 . . . . 5  |-  ( E! x  A ( F  |`  B ) x  <->  E. w { x  |  A
( F  |`  B ) x }  =  {
w } )
1816, 17sylibr 133 . . . 4  |-  ( z  e.  ( ( F  |`  B ) `  A
)  ->  E! x  A ( F  |`  B ) x )
19 euex 2005 . . . 4  |-  ( E! x  A ( F  |`  B ) x  ->  E. x  A ( F  |`  B ) x )
20 df-br 3898 . . . . . . . 8  |-  ( A ( F  |`  B ) x  <->  <. A ,  x >.  e.  ( F  |`  B ) )
21 df-res 4519 . . . . . . . . 9  |-  ( F  |`  B )  =  ( F  i^i  ( B  X.  _V ) )
2221eleq2i 2182 . . . . . . . 8  |-  ( <. A ,  x >.  e.  ( F  |`  B )  <->  <. A ,  x >.  e.  ( F  i^i  ( B  X.  _V ) ) )
2320, 22bitri 183 . . . . . . 7  |-  ( A ( F  |`  B ) x  <->  <. A ,  x >.  e.  ( F  i^i  ( B  X.  _V )
) )
24 elin 3227 . . . . . . . 8  |-  ( <. A ,  x >.  e.  ( F  i^i  ( B  X.  _V ) )  <-> 
( <. A ,  x >.  e.  F  /\  <. A ,  x >.  e.  ( B  X.  _V )
) )
2524simprbi 271 . . . . . . 7  |-  ( <. A ,  x >.  e.  ( F  i^i  ( B  X.  _V ) )  ->  <. A ,  x >.  e.  ( B  X.  _V ) )
2623, 25sylbi 120 . . . . . 6  |-  ( A ( F  |`  B ) x  ->  <. A ,  x >.  e.  ( B  X.  _V ) )
27 opelxp1 4541 . . . . . 6  |-  ( <. A ,  x >.  e.  ( B  X.  _V )  ->  A  e.  B
)
2826, 27syl 14 . . . . 5  |-  ( A ( F  |`  B ) x  ->  A  e.  B )
2928exlimiv 1560 . . . 4  |-  ( E. x  A ( F  |`  B ) x  ->  A  e.  B )
3018, 19, 293syl 17 . . 3  |-  ( z  e.  ( ( F  |`  B ) `  A
)  ->  A  e.  B )
3130con3i 604 . 2  |-  ( -.  A  e.  B  ->  -.  z  e.  (
( F  |`  B ) `
 A ) )
3231eq0rdv 3375 1  |-  ( -.  A  e.  B  -> 
( ( F  |`  B ) `  A
)  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    = wceq 1314   E.wex 1451    e. wcel 1463   [wsb 1718   E!weu 1975   {cab 2101   _Vcvv 2658    i^i cin 3038   (/)c0 3331   {csn 3495   <.cop 3498   U.cuni 3704   class class class wbr 3897    X. cxp 4505    |` cres 4509   iotacio 5054   ` cfv 5091
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-br 3898  df-opab 3958  df-xp 4513  df-res 4519  df-iota 5056  df-fv 5099
This theorem is referenced by: (None)
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