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Theorem 0fv 5449
Description: Function value of the empty set. (Contributed by Stefan O'Rear, 26-Nov-2014.)
Assertion
Ref Expression
0fv  |-  ( (/) `  A )  =  (/)

Proof of Theorem 0fv
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 df-fv 5126 . 2  |-  ( (/) `  A )  =  ( iota x A (/) x )
2 noel 3362 . . . . . 6  |-  -.  <. A ,  x >.  e.  (/)
3 df-br 3925 . . . . . 6  |-  ( A
(/) x  <->  <. A ,  x >.  e.  (/) )
42, 3mtbir 660 . . . . 5  |-  -.  A (/) x
54nex 1476 . . . 4  |-  -.  E. x  A (/) x
6 euex 2027 . . . 4  |-  ( E! x  A (/) x  ->  E. x  A (/) x )
75, 6mto 651 . . 3  |-  -.  E! x  A (/) x
8 iotanul 5098 . . 3  |-  ( -.  E! x  A (/) x  ->  ( iota x A (/) x )  =  (/) )
97, 8ax-mp 5 . 2  |-  ( iota
x A (/) x )  =  (/)
101, 9eqtri 2158 1  |-  ( (/) `  A )  =  (/)
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1331   E.wex 1468    e. wcel 1480   E!weu 1997   (/)c0 3358   <.cop 3525   class class class wbr 3924   iotacio 5081   ` cfv 5118
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-dif 3068  df-in 3072  df-ss 3079  df-nul 3359  df-sn 3528  df-uni 3732  df-br 3925  df-iota 5083  df-fv 5126
This theorem is referenced by:  strsl0  11996
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