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Theorem 0fv 5208
 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 4910 . 2 (∅‘𝐴) = (℩𝑥𝐴𝑥)
2 noel 3228 . . . . . 6 ¬ ⟨𝐴, 𝑥⟩ ∈ ∅
3 df-br 3765 . . . . . 6 (𝐴𝑥 ↔ ⟨𝐴, 𝑥⟩ ∈ ∅)
42, 3mtbir 596 . . . . 5 ¬ 𝐴𝑥
54nex 1389 . . . 4 ¬ ∃𝑥 𝐴𝑥
6 euex 1930 . . . 4 (∃!𝑥 𝐴𝑥 → ∃𝑥 𝐴𝑥)
75, 6mto 588 . . 3 ¬ ∃!𝑥 𝐴𝑥
8 iotanul 4882 . . 3 (¬ ∃!𝑥 𝐴𝑥 → (℩𝑥𝐴𝑥) = ∅)
97, 8ax-mp 7 . 2 (℩𝑥𝐴𝑥) = ∅
101, 9eqtri 2060 1 (∅‘𝐴) = ∅
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   = wceq 1243  ∃wex 1381   ∈ wcel 1393  ∃!weu 1900  ∅c0 3224  ⟨cop 3378   class class class wbr 3764  ℩cio 4865  ‘cfv 4902 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 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-dif 2920  df-in 2924  df-ss 2931  df-nul 3225  df-sn 3381  df-uni 3581  df-br 3765  df-iota 4867  df-fv 4910 This theorem is referenced by: (None)
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