Theorem 2nd0 6200

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		0ex 4157	. . 3 ⊢ ∅ ∈ V
2		2ndvalg 6198	. . 3 ⊢ (∅ ∈ V → (2nd ‘∅) = ∪ ran {∅})
3	1, 2	ax-mp 5	. 2 ⊢ (2nd ‘∅) = ∪ ran {∅}
4		dmsn0 5134	. . . 4 ⊢ dom {∅} = ∅
5		dm0rn0 4880	. . . 4 ⊢ (dom {∅} = ∅ ↔ ran {∅} = ∅)
6	4, 5	mpbi 145	. . 3 ⊢ ran {∅} = ∅
7	6	unieqi 3846	. 2 ⊢ ∪ ran {∅} = ∪ ∅
8		uni0 3863	. 2 ⊢ ∪ ∅ = ∅
9	3, 7, 8	3eqtri 2218	1 ⊢ (2nd ‘∅) = ∅

Syntax hints:   = wceq 1364   ∈ wcel 2164  Vcvv 2760  ∅c0 3447  {csn 3619  ∪ cuni 3836  dom cdm 4660  ran crn 4661  ‘cfv 5255  2^nd c2nd 6194

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-nul 4156  ax-pow 4204  ax-pr 4239  ax-un 4465

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2987  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-iota 5216  df-fun 5257  df-fv 5263  df-2nd 6196

This theorem is referenced by: (None)