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Theorem 2nd0 5772
 Description: The value of the second-member function at the empty set. (Contributed by NM, 23-Apr-2007.)
Assertion
Ref Expression
2nd0 (2nd ‘∅) = ∅

Proof of Theorem 2nd0
StepHypRef Expression
1 0ex 3884 . . 3 ∅ ∈ V
2 2ndvalg 5770 . . 3 (∅ ∈ V → (2nd ‘∅) = ran {∅})
31, 2ax-mp 7 . 2 (2nd ‘∅) = ran {∅}
4 dmsn0 4788 . . . 4 dom {∅} = ∅
5 dm0rn0 4552 . . . 4 (dom {∅} = ∅ ↔ ran {∅} = ∅)
64, 5mpbi 133 . . 3 ran {∅} = ∅
76unieqi 3590 . 2 ran {∅} =
8 uni0 3607 . 2 ∅ = ∅
93, 7, 83eqtri 2064 1 (2nd ‘∅) = ∅
 Colors of variables: wff set class Syntax hints:   = wceq 1243   ∈ wcel 1393  Vcvv 2557  ∅c0 3224  {csn 3375  ∪ cuni 3580  dom cdm 4345  ran crn 4346  ‘cfv 4902  2nd c2nd 5766 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-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-iota 4867  df-fun 4904  df-fv 4910  df-2nd 5768 This theorem is referenced by: (None)
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