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Theorem 0fv 5352
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 5036 . 2  |-  ( (/) `  A )  =  ( iota x A (/) x )
2 noel 3291 . . . . . 6  |-  -.  <. A ,  x >.  e.  (/)
3 df-br 3852 . . . . . 6  |-  ( A
(/) x  <->  <. A ,  x >.  e.  (/) )
42, 3mtbir 632 . . . . 5  |-  -.  A (/) x
54nex 1435 . . . 4  |-  -.  E. x  A (/) x
6 euex 1979 . . . 4  |-  ( E! x  A (/) x  ->  E. x  A (/) x )
75, 6mto 624 . . 3  |-  -.  E! x  A (/) x
8 iotanul 5008 . . 3  |-  ( -.  E! x  A (/) x  ->  ( iota x A (/) x )  =  (/) )
97, 8ax-mp 7 . 2  |-  ( iota
x A (/) x )  =  (/)
101, 9eqtri 2109 1  |-  ( (/) `  A )  =  (/)
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1290   E.wex 1427    e. wcel 1439   E!weu 1949   (/)c0 3287   <.cop 3453   class class class wbr 3851   iotacio 4991   ` cfv 5028
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071
This theorem depends on definitions:  df-bi 116  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2622  df-dif 3002  df-in 3006  df-ss 3013  df-nul 3288  df-sn 3456  df-uni 3660  df-br 3852  df-iota 4993  df-fv 5036
This theorem is referenced by:  strsl0  11596
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