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 6185	. . . 4 ⊢ dom {∅} = ∅
3		dm0rn0 5891	. . . 4 ⊢ (dom {∅} = ∅ ↔ ran {∅} = ∅)
4	2, 3	mpbi 230	. . 3 ⊢ ran {∅} = ∅
5	4	unieqi 4886	. 2 ⊢ ∪ ran {∅} = ∪ ∅
6		uni0 4902	. 2 ⊢ ∪ ∅ = ∅
7	1, 5, 6	3eqtri 2757	1 ⊢ (2nd ‘∅) = ∅

Syntax hints:   = wceq 1540  ∅c0 4299  {csn 4592  ∪ cuni 4874  dom cdm 5641  ran crn 5642  ‘cfv 6514  2^nd c2nd 7970

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-iota 6467  df-fun 6516  df-fv 6522  df-2nd 7972

This theorem is referenced by:  smfval  30541  fucofvalne  49318  reldmprcof2  49375