Theorem 2nd0 7942

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 7938	. 2 ⊢ (2nd ‘∅) = ∪ ran {∅}
2		dmsn0 6168	. . . 4 ⊢ dom {∅} = ∅
3		dm0rn0 5874	. . . 4 ⊢ (dom {∅} = ∅ ↔ ran {∅} = ∅)
4	2, 3	mpbi 230	. . 3 ⊢ ran {∅} = ∅
5	4	unieqi 4876	. 2 ⊢ ∪ ran {∅} = ∪ ∅
6		uni0 4892	. 2 ⊢ ∪ ∅ = ∅
7	1, 5, 6	3eqtri 2764	1 ⊢ (2nd ‘∅) = ∅

Syntax hints:   = wceq 1542  ∅c0 4286  {csn 4581  ∪ cuni 4864  dom cdm 5625  ran crn 5626  ‘cfv 6493  2^nd c2nd 7934

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5242  ax-nul 5252  ax-pr 5378  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3401  df-v 3443  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4287  df-if 4481  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-iota 6449  df-fun 6495  df-fv 6501  df-2nd 7936

This theorem is referenced by:  smfval  30663  fucofvalne  49606  reldmprcof2  49663