Theorem 2nd0 8002

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 7998	. 2 ⊢ (2nd ‘∅) = ∪ ran {∅}
2		dmsn0 6215	. . . 4 ⊢ dom {∅} = ∅
3		dm0rn0 5919	. . . 4 ⊢ (dom {∅} = ∅ ↔ ran {∅} = ∅)
4	2, 3	mpbi 233	. . 3 ⊢ ran {∅} = ∅
5	4	unieqi 4889	. 2 ⊢ ∪ ran {∅} = ∪ ∅
6		uni0 4906	. 2 ⊢ ∪ ∅ = ∅
7	1, 5, 6	3eqtri 2793	1 ⊢ (2nd ‘∅) = ∅

This proof depends on syntax axioms:   = wceq 1570  ∅c0 4289  {csn 4594  ∪ cuni 4877  dom cdm 5666  ran crn 5667  ‘cfv 6543  2^nd c2nd 7994

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-sep 5262  ax-nul 5274  ax-pr 5409  ax-un 7745

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-opab 5179  df-mpt 5198  df-id 5561  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-iota 6499  df-fun 6545  df-fv 6551  df-2nd 7996

This theorem is used by:  smfval  30994  fucofvalne  50143  reldmprcof2  50200