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Theorem nfvres 5238
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 4940 . . . . . . . . . 10  |-  ( ( F  |`  B ) `  A )  =  ( iota x A ( F  |`  B )
x )
2 df-iota 4897 . . . . . . . . . 10  |-  ( iota
x A ( F  |`  B ) x )  =  U. { y  |  { x  |  A ( F  |`  B ) x }  =  { y } }
31, 2eqtri 2102 . . . . . . . . 9  |-  ( ( F  |`  B ) `  A )  =  U. { y  |  {
x  |  A ( F  |`  B )
x }  =  {
y } }
43eleq2i 2146 . . . . . . . 8  |-  ( z  e.  ( ( F  |`  B ) `  A
)  <->  z  e.  U. { y  |  {
x  |  A ( F  |`  B )
x }  =  {
y } } )
5 eluni 3612 . . . . . . . 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 1550 . . . . . . 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 2069 . . . . . . . 8  |-  ( w  e.  { y  |  { x  |  A
( F  |`  B ) x }  =  {
y } }  <->  [ w  /  y ] {
x  |  A ( F  |`  B )
x }  =  {
y } )
10 nfv 1462 . . . . . . . . 9  |-  F/ y { x  |  A
( F  |`  B ) x }  =  {
w }
11 sneq 3417 . . . . . . . . . 10  |-  ( y  =  w  ->  { y }  =  { w } )
1211eqeq2d 2093 . . . . . . . . 9  |-  ( y  =  w  ->  ( { x  |  A
( F  |`  B ) x }  =  {
y }  <->  { x  |  A ( F  |`  B ) x }  =  { w } ) )
1310, 12sbie 1715 . . . . . . . 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 1537 . . . . . 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 3469 . . . . 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 1972 . . . 4  |-  ( E! x  A ( F  |`  B ) x  ->  E. x  A ( F  |`  B ) x )
20 df-br 3794 . . . . . . . 8  |-  ( A ( F  |`  B ) x  <->  <. A ,  x >.  e.  ( F  |`  B ) )
21 df-res 4383 . . . . . . . . 9  |-  ( F  |`  B )  =  ( F  i^i  ( B  X.  _V ) )
2221eleq2i 2146 . . . . . . . 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 3156 . . . . . . . 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 4403 . . . . . 6  |-  ( <. A ,  x >.  e.  ( B  X.  _V )  ->  A  e.  B
)
2826, 27syl 14 . . . . 5  |-  ( A ( F  |`  B ) x  ->  A  e.  B )
2928exlimiv 1530 . . . 4  |-  ( E. x  A ( F  |`  B ) x  ->  A  e.  B )
3018, 19, 293syl 17 . . 3  |-  ( z  e.  ( ( F  |`  B ) `  A
)  ->  A  e.  B )
3130con3i 595 . 2  |-  ( -.  A  e.  B  ->  -.  z  e.  (
( F  |`  B ) `
 A ) )
3231eq0rdv 3295 1  |-  ( -.  A  e.  B  -> 
( ( F  |`  B ) `  A
)  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    = wceq 1285   E.wex 1422    e. wcel 1434   [wsb 1686   E!weu 1942   {cab 2068   _Vcvv 2602    i^i cin 2973   (/)c0 3258   {csn 3406   <.cop 3409   U.cuni 3609   class class class wbr 3793    X. cxp 4369    |` cres 4373   iotacio 4895   ` cfv 4932
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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3259  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-br 3794  df-opab 3848  df-xp 4377  df-res 4383  df-iota 4897  df-fv 4940
This theorem is referenced by: (None)
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