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Theorem 2nd0 7454
 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 2ndval 7450 . 2 (2nd ‘∅) = ran {∅}
2 dmsn0 5858 . . . 4 dom {∅} = ∅
3 dm0rn0 5589 . . . 4 (dom {∅} = ∅ ↔ ran {∅} = ∅)
42, 3mpbi 222 . . 3 ran {∅} = ∅
54unieqi 4682 . 2 ran {∅} =
6 uni0 4702 . 2 ∅ = ∅
71, 5, 63eqtri 2806 1 (2nd ‘∅) = ∅
 Colors of variables: wff setvar class Syntax hints:   = wceq 1601  ∅c0 4141  {csn 4398  ∪ cuni 4673  dom cdm 5357  ran crn 5358  ‘cfv 6137  2nd c2nd 7446 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5019  ax-nul 5027  ax-pow 5079  ax-pr 5140  ax-un 7228 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4674  df-br 4889  df-opab 4951  df-mpt 4968  df-id 5263  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-iota 6101  df-fun 6139  df-fv 6145  df-2nd 7448 This theorem is referenced by:  smfval  28036
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