Theorem 2nd0 7981

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 7977	. 2 ⊢ (2nd ‘∅) = ∪ ran {∅}
2		dmsn0 6200	. . . 4 ⊢ dom {∅} = ∅
3		dm0rn0 5905	. . . 4 ⊢ (dom {∅} = ∅ ↔ ran {∅} = ∅)
4	2, 3	mpbi 233	. . 3 ⊢ ran {∅} = ∅
5	4	unieqi 4880	. 2 ⊢ ∪ ran {∅} = ∪ ∅
6		uni0 4897	. 2 ⊢ ∪ ∅ = ∅
7	1, 5, 6	3eqtri 2792	1 ⊢ (2nd ‘∅) = ∅

Syntax hints:   = wceq 1563  ∅c0 4288  {csn 4585  ∪ cuni 4868  dom cdm 5652  ran crn 5653  ‘cfv 6525  2^nd c2nd 7973

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5251  ax-nul 5261  ax-pr 5395  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-br 5106  df-opab 5168  df-mpt 5187  df-id 5547  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-iota 6481  df-fun 6527  df-fv 6533  df-2nd 7975

This theorem is referenced by:  smfval  30866  fucofvalne  49954  reldmprcof2  50011