Theorem 2nd0 6113

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 4109	. . 3 ⊢ ∅ ∈ V
2		2ndvalg 6111	. . 3 ⊢ (∅ ∈ V → (2nd ‘∅) = ∪ ran {∅})
3	1, 2	ax-mp 5	. 2 ⊢ (2nd ‘∅) = ∪ ran {∅}
4		dmsn0 5071	. . . 4 ⊢ dom {∅} = ∅
5		dm0rn0 4821	. . . 4 ⊢ (dom {∅} = ∅ ↔ ran {∅} = ∅)
6	4, 5	mpbi 144	. . 3 ⊢ ran {∅} = ∅
7	6	unieqi 3799	. 2 ⊢ ∪ ran {∅} = ∪ ∅
8		uni0 3816	. 2 ⊢ ∪ ∅ = ∅
9	3, 7, 8	3eqtri 2190	1 ⊢ (2nd ‘∅) = ∅

Syntax hints:   = wceq 1343   ∈ wcel 2136  Vcvv 2726  ∅c0 3409  {csn 3576  ∪ cuni 3789  dom cdm 4604  ran crn 4605  ‘cfv 5188  2^nd c2nd 6107

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-v 2728  df-sbc 2952  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-iota 5153  df-fun 5190  df-fv 5196  df-2nd 6109

This theorem is referenced by: (None)