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Theorem 0fv 5151
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 4853 . 2 (∅‘A) = (℩xAx)
2 noel 3222 . . . . . 6 ¬ ⟨A, x
3 df-br 3756 . . . . . 6 (Ax ↔ ⟨A, x ∅)
42, 3mtbir 595 . . . . 5 ¬ Ax
54nex 1386 . . . 4 ¬ x Ax
6 euex 1927 . . . 4 (∃!x Axx Ax)
75, 6mto 587 . . 3 ¬ ∃!x Ax
8 iotanul 4825 . . 3 ∃!x Ax → (℩xAx) = ∅)
97, 8ax-mp 7 . 2 (℩xAx) = ∅
101, 9eqtri 2057 1 (∅‘A) = ∅
Colors of variables: wff set class
Syntax hints:  ¬ wn 3   = wceq 1242  wex 1378   wcel 1390  ∃!weu 1897  c0 3218  cop 3370   class class class wbr 3755  cio 4808  cfv 4845
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-dif 2914  df-in 2918  df-ss 2925  df-nul 3219  df-sn 3373  df-uni 3572  df-br 3756  df-iota 4810  df-fv 4853
This theorem is referenced by: (None)
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