Theorem 2nd0 7945

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 7941	. 2 ⊢ (2nd ‘∅) = ∪ ran {∅}
2		dmsn0 6167	. . . 4 ⊢ dom {∅} = ∅
3		dm0rn0 5873	. . . 4 ⊢ (dom {∅} = ∅ ↔ ran {∅} = ∅)
4	2, 3	mpbi 231	. . 3 ⊢ ran {∅} = ∅
5	4	unieqi 4857	. 2 ⊢ ∪ ran {∅} = ∪ ∅
6		uni0 4873	. 2 ⊢ ∪ ∅ = ∅
7	1, 5, 6	3eqtri 2767	1 ⊢ (2nd ‘∅) = ∅

Syntax hints:   = wceq 1547  ∅c0 4268  {csn 4562  ∪ cuni 4845  dom cdm 5625  ran crn 5626  ‘cfv 6492  2^nd c2nd 7937

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-nul 5235  ax-pr 5369  ax-un 7685

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-br 5080  df-opab 5142  df-mpt 5161  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-iota 6448  df-fun 6494  df-fv 6500  df-2nd 7939

This theorem is referenced by:  smfval  30701  fucofvalne  49822  reldmprcof2  49879