Theorem 2nd0 6124

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 4116	. . 3 ⊢ ∅ ∈ V
2		2ndvalg 6122	. . 3 ⊢ (∅ ∈ V → (2nd ‘∅) = ∪ ran {∅})
3	1, 2	ax-mp 5	. 2 ⊢ (2nd ‘∅) = ∪ ran {∅}
4		dmsn0 5078	. . . 4 ⊢ dom {∅} = ∅
5		dm0rn0 4828	. . . 4 ⊢ (dom {∅} = ∅ ↔ ran {∅} = ∅)
6	4, 5	mpbi 144	. . 3 ⊢ ran {∅} = ∅
7	6	unieqi 3806	. 2 ⊢ ∪ ran {∅} = ∪ ∅
8		uni0 3823	. 2 ⊢ ∪ ∅ = ∅
9	3, 7, 8	3eqtri 2195	1 ⊢ (2nd ‘∅) = ∅

Syntax hints:   = wceq 1348   ∈ wcel 2141  Vcvv 2730  ∅c0 3414  {csn 3583  ∪ cuni 3796  dom cdm 4611  ran crn 4612  ‘cfv 5198  2^nd c2nd 6118

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-iota 5160  df-fun 5200  df-fv 5206  df-2nd 6120

This theorem is referenced by: (None)