Theorem 2nd0 6261

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 4190	. . 3 ⊢ ∅ ∈ V
2		2ndvalg 6259	. . 3 ⊢ (∅ ∈ V → (2nd ‘∅) = ∪ ran {∅})
3	1, 2	ax-mp 5	. 2 ⊢ (2nd ‘∅) = ∪ ran {∅}
4		dmsn0 5172	. . . 4 ⊢ dom {∅} = ∅
5		dm0rn0 4917	. . . 4 ⊢ (dom {∅} = ∅ ↔ ran {∅} = ∅)
6	4, 5	mpbi 145	. . 3 ⊢ ran {∅} = ∅
7	6	unieqi 3877	. 2 ⊢ ∪ ran {∅} = ∪ ∅
8		uni0 3894	. 2 ⊢ ∪ ∅ = ∅
9	3, 7, 8	3eqtri 2234	1 ⊢ (2nd ‘∅) = ∅

Syntax hints:   = wceq 1375   ∈ wcel 2180  Vcvv 2779  ∅c0 3471  {csn 3646  ∪ cuni 3867  dom cdm 4696  ran crn 4697  ‘cfv 5294  2^nd c2nd 6255

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 617  ax-in2 618  ax-io 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-13 2182  ax-14 2183  ax-ext 2191  ax-sep 4181  ax-nul 4189  ax-pow 4237  ax-pr 4272  ax-un 4501

This theorem depends on definitions:  df-bi 117  df-3an 985  df-tru 1378  df-fal 1381  df-nf 1487  df-sb 1789  df-eu 2060  df-mo 2061  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-ne 2381  df-ral 2493  df-rex 2494  df-v 2781  df-sbc 3009  df-dif 3179  df-un 3181  df-in 3183  df-ss 3190  df-nul 3472  df-pw 3631  df-sn 3652  df-pr 3653  df-op 3655  df-uni 3868  df-br 4063  df-opab 4125  df-mpt 4126  df-id 4361  df-xp 4702  df-rel 4703  df-cnv 4704  df-co 4705  df-dm 4706  df-rn 4707  df-iota 5254  df-fun 5296  df-fv 5302  df-2nd 6257

This theorem is referenced by: (None)