Theorem 2nd0 7995

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 7991	. 2 ⊢ (2nd ‘∅) = ∪ ran {∅}
2		dmsn0 6198	. . . 4 ⊢ dom {∅} = ∅
3		dm0rn0 5904	. . . 4 ⊢ (dom {∅} = ∅ ↔ ran {∅} = ∅)
4	2, 3	mpbi 230	. . 3 ⊢ ran {∅} = ∅
5	4	unieqi 4895	. 2 ⊢ ∪ ran {∅} = ∪ ∅
6		uni0 4911	. 2 ⊢ ∪ ∅ = ∅
7	1, 5, 6	3eqtri 2762	1 ⊢ (2nd ‘∅) = ∅

Syntax hints:   = wceq 1540  ∅c0 4308  {csn 4601  ∪ cuni 4883  dom cdm 5654  ran crn 5655  ‘cfv 6531  2^nd c2nd 7987

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402  ax-un 7729

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-iota 6484  df-fun 6533  df-fv 6539  df-2nd 7989

This theorem is referenced by:  smfval  30586  fucofvalne  49236  reldmprcof2  49292