Theorem 2nd0 7997

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 7993	. 2 ⊢ (2nd ‘∅) = ∪ ran {∅}
2		dmsn0 6203	. . . 4 ⊢ dom {∅} = ∅
3		dm0rn0 5906	. . . 4 ⊢ (dom {∅} = ∅ ↔ ran {∅} = ∅)
4	2, 3	mpbi 233	. . 3 ⊢ ran {∅} = ∅
5	4	unieqi 4879	. 2 ⊢ ∪ ran {∅} = ∪ ∅
6		uni0 4896	. 2 ⊢ ∪ ∅ = ∅
7	1, 5, 6	3eqtri 2788	1 ⊢ (2nd ‘∅) = ∅

This proof depends on syntax axioms:   = wceq 1570  ∅c0 4279  {csn 4584  ∪ cuni 4867  dom cdm 5651  ran crn 5652  ‘cfv 6531  2^nd c2nd 7989

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pr 5391  ax-un 7740

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-iota 6487  df-fun 6533  df-fv 6539  df-2nd 7991

This theorem is used by:  smfval  31189  fucofvalne  50377  reldmprcof2  50434