Theorem 2nd0 6338

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 4236	. . 3 ⊢ ∅ ∈ V
2		2ndvalg 6336	. . 3 ⊢ (∅ ∈ V → (2nd ‘∅) = ∪ ran {∅})
3	1, 2	ax-mp 5	. 2 ⊢ (2nd ‘∅) = ∪ ran {∅}
4		dmsn0 5229	. . . 4 ⊢ dom {∅} = ∅
5		dm0rn0 4972	. . . 4 ⊢ (dom {∅} = ∅ ↔ ran {∅} = ∅)
6	4, 5	mpbi 145	. . 3 ⊢ ran {∅} = ∅
7	6	unieqi 3923	. 2 ⊢ ∪ ran {∅} = ∪ ∅
8		uni0 3940	. 2 ⊢ ∪ ∅ = ∅
9	3, 7, 8	3eqtri 2257	1 ⊢ (2nd ‘∅) = ∅

Syntax hints:   = wceq 1398   ∈ wcel 2203  Vcvv 2812  ∅c0 3507  {csn 3688  ∪ cuni 3913  dom cdm 4748  ran crn 4749  ‘cfv 5351  2^nd c2nd 6332

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4227  ax-nul 4235  ax-pow 4286  ax-pr 4321  ax-un 4553

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-v 2814  df-sbc 3042  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-br 4109  df-opab 4171  df-mpt 4172  df-id 4413  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-iota 5311  df-fun 5353  df-fv 5359  df-2nd 6334

This theorem is referenced by: (None)