Theorem 2nd0 7454

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 7450	. 2 ⊢ (2nd ‘∅) = ∪ ran {∅}
2		dmsn0 5858	. . . 4 ⊢ dom {∅} = ∅
3		dm0rn0 5589	. . . 4 ⊢ (dom {∅} = ∅ ↔ ran {∅} = ∅)
4	2, 3	mpbi 222	. . 3 ⊢ ran {∅} = ∅
5	4	unieqi 4682	. 2 ⊢ ∪ ran {∅} = ∪ ∅
6		uni0 4702	. 2 ⊢ ∪ ∅ = ∅
7	1, 5, 6	3eqtri 2806	1 ⊢ (2nd ‘∅) = ∅

Syntax hints:   = wceq 1601  ∅c0 4141  {csn 4398  ∪ cuni 4673  dom cdm 5357  ran crn 5358  ‘cfv 6137  2^nd c2nd 7446

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5019  ax-nul 5027  ax-pow 5079  ax-pr 5140  ax-un 7228

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4674  df-br 4889  df-opab 4951  df-mpt 4968  df-id 5263  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-iota 6101  df-fun 6139  df-fv 6145  df-2nd 7448

This theorem is referenced by:  smfval  28036