Theorem 2nd0 7978

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

Step	Hyp	Ref	Expression
1		2ndval 7974	. 2 ⊢ (2nd ‘∅) = ∪ ran {∅}
2		dmsn0 6205	. . . 4 ⊢ dom {∅} = ∅
3		dm0rn0 5922	. . . 4 ⊢ (dom {∅} = ∅ ↔ ran {∅} = ∅)
4	2, 3	mpbi 229	. . 3 ⊢ ran {∅} = ∅
5	4	unieqi 4920	. 2 ⊢ ∪ ran {∅} = ∪ ∅
6		uni0 4938	. 2 ⊢ ∪ ∅ = ∅
7	1, 5, 6	3eqtri 2764	1 ⊢ (2nd ‘∅) = ∅

Syntax hints:   = wceq 1541  ∅c0 4321  {csn 4627  ∪ cuni 4907  dom cdm 5675  ran crn 5676  ‘cfv 6540  2^nd c2nd 7970

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7721

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-iota 6492  df-fun 6542  df-fv 6548  df-2nd 7972

This theorem is referenced by:  smfval  29845