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Theorem 0fv 5129
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 4833 . 2 (∅‘A) = (℩xAx)
2 noel 3201 . . . . . 6 ¬ ⟨A, x
3 df-br 3735 . . . . . 6 (Ax ↔ ⟨A, x ∅)
42, 3mtbir 583 . . . . 5 ¬ Ax
54nex 1366 . . . 4 ¬ x Ax
6 euex 1908 . . . 4 (∃!x Axx Ax)
75, 6mto 575 . . 3 ¬ ∃!x Ax
8 iotanul 4805 . . 3 ∃!x Ax → (℩xAx) = ∅)
97, 8ax-mp 7 . 2 (℩xAx) = ∅
101, 9eqtri 2038 1 (∅‘A) = ∅
Colors of variables: wff set class
Syntax hints:  ¬ wn 3   = wceq 1226  wex 1358   wcel 1370  ∃!weu 1878  c0 3197  cop 3349   class class class wbr 3734  cio 4788  cfv 4825
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 532  ax-in2 533  ax-io 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000
This theorem depends on definitions:  df-bi 110  df-tru 1229  df-fal 1232  df-nf 1326  df-sb 1624  df-eu 1881  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ral 2285  df-rex 2286  df-v 2533  df-dif 2893  df-in 2897  df-ss 2904  df-nul 3198  df-sn 3352  df-uni 3551  df-br 3735  df-iota 4790  df-fv 4833
This theorem is referenced by:  rdgi0g  5882  rdg0  5891
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