Theorem 2nd0 7994

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 7990	. 2 ⊢ (2nd ‘∅) = ∪ ran {∅}
2		dmsn0 6212	. . . 4 ⊢ dom {∅} = ∅
3		dm0rn0 5916	. . . 4 ⊢ (dom {∅} = ∅ ↔ ran {∅} = ∅)
4	2, 3	mpbi 233	. . 3 ⊢ ran {∅} = ∅
5	4	unieqi 4885	. 2 ⊢ ∪ ran {∅} = ∪ ∅
6		uni0 4902	. 2 ⊢ ∪ ∅ = ∅
7	1, 5, 6	3eqtri 2790	1 ⊢ (2nd ‘∅) = ∅

Syntax hints:   = wceq 1570  ∅c0 4287  {csn 4590  ∪ cuni 4873  dom cdm 5663  ran crn 5664  ‘cfv 6538  2^nd c2nd 7986

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-nul 5270  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-iota 6494  df-fun 6540  df-fv 6546  df-2nd 7988

This theorem is referenced by:  smfval  30935  fucofvalne  50080  reldmprcof2  50137