Theorem 2nd0 5772

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 3884	. . 3 ⊢ ∅ ∈ V
2		2ndvalg 5770	. . 3 ⊢ (∅ ∈ V → (2nd ‘∅) = ∪ ran {∅})
3	1, 2	ax-mp 7	. 2 ⊢ (2nd ‘∅) = ∪ ran {∅}
4		dmsn0 4788	. . . 4 ⊢ dom {∅} = ∅
5		dm0rn0 4552	. . . 4 ⊢ (dom {∅} = ∅ ↔ ran {∅} = ∅)
6	4, 5	mpbi 133	. . 3 ⊢ ran {∅} = ∅
7	6	unieqi 3590	. 2 ⊢ ∪ ran {∅} = ∪ ∅
8		uni0 3607	. 2 ⊢ ∪ ∅ = ∅
9	3, 7, 8	3eqtri 2064	1 ⊢ (2nd ‘∅) = ∅

Syntax hints:   = wceq 1243   ∈ wcel 1393  Vcvv 2557  ∅c0 3224  {csn 3375  ∪ cuni 3580  dom cdm 4345  ran crn 4346  ‘cfv 4902  2^nd c2nd 5766

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-iota 4867  df-fun 4904  df-fv 4910  df-2nd 5768

This theorem is referenced by: (None)