Theorem 1st0 7975

Description: The value of the first-member function at the empty set. (Contributed by NM, 23-Apr-2007.)

Assertion

Ref	Expression
1st0	⊢ (1st ‘∅) = ∅

Proof of Theorem 1st0

Step	Hyp	Ref	Expression
1		1stval 7971	. 2 ⊢ (1st ‘∅) = ∪ dom {∅}
2		dmsn0 6199	. . 3 ⊢ dom {∅} = ∅
3	2	unieqi 4912	. 2 ⊢ ∪ dom {∅} = ∪ ∅
4		uni0 4930	. 2 ⊢ ∪ ∅ = ∅
5	1, 3, 4	3eqtri 2756	1 ⊢ (1st ‘∅) = ∅

Syntax hints:   = wceq 1533  ∅c0 4315  {csn 4621  ∪ cuni 4900  dom cdm 5667  ‘cfv 6534  1^st c1st 7967

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418  ax-un 7719

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-iota 6486  df-fun 6536  df-fv 6542  df-1st 7969

This theorem is referenced by:  vafval  30328