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Theorem 0fv 5236
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 4938 . 2  |-  ( (/) `  A )  =  ( iota x A (/) x )
2 noel 3256 . . . . . 6  |-  -.  <. A ,  x >.  e.  (/)
3 df-br 3793 . . . . . 6  |-  ( A
(/) x  <->  <. A ,  x >.  e.  (/) )
42, 3mtbir 606 . . . . 5  |-  -.  A (/) x
54nex 1405 . . . 4  |-  -.  E. x  A (/) x
6 euex 1946 . . . 4  |-  ( E! x  A (/) x  ->  E. x  A (/) x )
75, 6mto 598 . . 3  |-  -.  E! x  A (/) x
8 iotanul 4910 . . 3  |-  ( -.  E! x  A (/) x  ->  ( iota x A (/) x )  =  (/) )
97, 8ax-mp 7 . 2  |-  ( iota
x A (/) x )  =  (/)
101, 9eqtri 2076 1  |-  ( (/) `  A )  =  (/)
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1259   E.wex 1397    e. wcel 1409   E!weu 1916   (/)c0 3252   <.cop 3406   class class class wbr 3792   iotacio 4893   ` cfv 4930
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-dif 2948  df-in 2952  df-ss 2959  df-nul 3253  df-sn 3409  df-uni 3609  df-br 3793  df-iota 4895  df-fv 4938
This theorem is referenced by: (None)
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