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Theorem 0fv 5323
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 5010 . 2 (∅‘𝐴) = (℩𝑥𝐴𝑥)
2 noel 3288 . . . . . 6 ¬ ⟨𝐴, 𝑥⟩ ∈ ∅
3 df-br 3838 . . . . . 6 (𝐴𝑥 ↔ ⟨𝐴, 𝑥⟩ ∈ ∅)
42, 3mtbir 631 . . . . 5 ¬ 𝐴𝑥
54nex 1434 . . . 4 ¬ ∃𝑥 𝐴𝑥
6 euex 1978 . . . 4 (∃!𝑥 𝐴𝑥 → ∃𝑥 𝐴𝑥)
75, 6mto 623 . . 3 ¬ ∃!𝑥 𝐴𝑥
8 iotanul 4982 . . 3 (¬ ∃!𝑥 𝐴𝑥 → (℩𝑥𝐴𝑥) = ∅)
97, 8ax-mp 7 . 2 (℩𝑥𝐴𝑥) = ∅
101, 9eqtri 2108 1 (∅‘𝐴) = ∅
Colors of variables: wff set class
Syntax hints:  ¬ wn 3   = wceq 1289  wex 1426  wcel 1438  ∃!weu 1948  c0 3284  cop 3444   class class class wbr 3837  cio 4965  cfv 5002
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-dif 2999  df-in 3003  df-ss 3010  df-nul 3285  df-sn 3447  df-uni 3649  df-br 3838  df-iota 4967  df-fv 5010
This theorem is referenced by: (None)
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