Theorem 2nd0 7994

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		2ndval 7990	. 2 ⊢ (2nd ‘∅) = ∪ ran {∅}
2		dmsn0 6207	. . . 4 ⊢ dom {∅} = ∅
3		dm0rn0 5921	. . . 4 ⊢ (dom {∅} = ∅ ↔ ran {∅} = ∅)
4	2, 3	mpbi 229	. . 3 ⊢ ran {∅} = ∅
5	4	unieqi 4915	. 2 ⊢ ∪ ran {∅} = ∪ ∅
6		uni0 4933	. 2 ⊢ ∪ ∅ = ∅
7	1, 5, 6	3eqtri 2760	1 ⊢ (2nd ‘∅) = ∅

Syntax hints:   = wceq 1534  ∅c0 4318  {csn 4624  ∪ cuni 4903  dom cdm 5672  ran crn 5673  ‘cfv 6542  2^nd c2nd 7986

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5293  ax-nul 5300  ax-pr 5423  ax-un 7734

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3472  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-iota 6494  df-fun 6544  df-fv 6550  df-2nd 7988

This theorem is referenced by:  smfval  30408