Theorem 2nd0 7066

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 7062	. 2 ⊢ (2nd ‘∅) = ∪ ran {∅}
2		dmsn0 5520	. . . 4 ⊢ dom {∅} = ∅
3		dm0rn0 5263	. . . 4 ⊢ (dom {∅} = ∅ ↔ ran {∅} = ∅)
4	2, 3	mpbi 219	. . 3 ⊢ ran {∅} = ∅
5	4	unieqi 4381	. 2 ⊢ ∪ ran {∅} = ∪ ∅
6		uni0 4401	. 2 ⊢ ∪ ∅ = ∅
7	1, 5, 6	3eqtri 2636	1 ⊢ (2nd ‘∅) = ∅

Syntax hints:   = wceq 1475  ∅c0 3874  {csn 4125  ∪ cuni 4372  dom cdm 5038  ran crn 5039  ‘cfv 5804  2^nd c2nd 7058

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fv 5812  df-2nd 7060

This theorem is referenced by:  smfval  26844