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Theorem 0fv 5235
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 4937 . 2 (∅‘𝐴) = (℩𝑥𝐴𝑥)
2 noel 3255 . . . . . 6 ¬ ⟨𝐴, 𝑥⟩ ∈ ∅
3 df-br 3792 . . . . . 6 (𝐴𝑥 ↔ ⟨𝐴, 𝑥⟩ ∈ ∅)
42, 3mtbir 606 . . . . 5 ¬ 𝐴𝑥
54nex 1405 . . . 4 ¬ ∃𝑥 𝐴𝑥
6 euex 1946 . . . 4 (∃!𝑥 𝐴𝑥 → ∃𝑥 𝐴𝑥)
75, 6mto 598 . . 3 ¬ ∃!𝑥 𝐴𝑥
8 iotanul 4909 . . 3 (¬ ∃!𝑥 𝐴𝑥 → (℩𝑥𝐴𝑥) = ∅)
97, 8ax-mp 7 . 2 (℩𝑥𝐴𝑥) = ∅
101, 9eqtri 2076 1 (∅‘𝐴) = ∅
Colors of variables: wff set class
Syntax hints:  ¬ wn 3   = wceq 1259  wex 1397  wcel 1409  ∃!weu 1916  c0 3251  cop 3405   class class class wbr 3791  cio 4892  cfv 4929
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 2947  df-in 2951  df-ss 2958  df-nul 3252  df-sn 3408  df-uni 3608  df-br 3792  df-iota 4894  df-fv 4937
This theorem is referenced by: (None)
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