Theorem 2nd0 5792

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 3905	. . 3 ⊢ ∅ ∈ V
2		2ndvalg 5790	. . 3 ⊢ (∅ ∈ V → (2nd ‘∅) = ∪ ran {∅})
3	1, 2	ax-mp 7	. 2 ⊢ (2nd ‘∅) = ∪ ran {∅}
4		dmsn0 4808	. . . 4 ⊢ dom {∅} = ∅
5		dm0rn0 4570	. . . 4 ⊢ (dom {∅} = ∅ ↔ ran {∅} = ∅)
6	4, 5	mpbi 143	. . 3 ⊢ ran {∅} = ∅
7	6	unieqi 3611	. 2 ⊢ ∪ ran {∅} = ∪ ∅
8		uni0 3628	. 2 ⊢ ∪ ∅ = ∅
9	3, 7, 8	3eqtri 2105	1 ⊢ (2nd ‘∅) = ∅

Syntax hints:   = wceq 1284   ∈ wcel 1433  Vcvv 2601  ∅c0 3251  {csn 3398  ∪ cuni 3601  dom cdm 4363  ran crn 4364  ‘cfv 4922  2^nd c2nd 5786

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-nul 3904  ax-pow 3948  ax-pr 3964  ax-un 4188

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-ral 2353  df-rex 2354  df-v 2603  df-sbc 2816  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-mpt 3841  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-iota 4887  df-fun 4924  df-fv 4930  df-2nd 5788

This theorem is referenced by: (None)