Theorem 2nd0 8032

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 8028	. 2 ⊢ (2nd ‘∅) = ∪ ran {∅}
2		dmsn0 6223	. . . 4 ⊢ dom {∅} = ∅
3		dm0rn0 5913	. . . 4 ⊢ (dom {∅} = ∅ ↔ ran {∅} = ∅)
4	2, 3	mpbi 233	. . 3 ⊢ ran {∅} = ∅
5	4	unieqi 4882	. 2 ⊢ ∪ ran {∅} = ∪ ∅
6		uni0 4899	. 2 ⊢ ∪ ∅ = ∅
7	1, 5, 6	3eqtri 2790	1 ⊢ (2nd ‘∅) = ∅

This proof depends on syntax axioms:   = wceq 1570  ∅c0 4281  {csn 4588  ∪ cuni 4870  dom cdm 5655  ran crn 5656  ‘cfv 6558  2^nd c2nd 8024

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pr 5395  ax-un 7775

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-iota 6512  df-fun 6560  df-fv 6566  df-2nd 8026

This theorem is used by:  smfval  31237  fucofvalne  50652  reldmprcof2  50722