Theorem 1st0 7963

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 7959	. 2 ⊢ (1st ‘∅) = ∪ dom {∅}
2		dmsn0 6197	. . 3 ⊢ dom {∅} = ∅
3	2	unieqi 4914	. 2 ⊢ ∪ dom {∅} = ∪ ∅
4		uni0 4932	. 2 ⊢ ∪ ∅ = ∅
5	1, 3, 4	3eqtri 2763	1 ⊢ (1st ‘∅) = ∅

Syntax hints:   = wceq 1541  ∅c0 4318  {csn 4622  ∪ cuni 4901  dom cdm 5669  ‘cfv 6532  1^st c1st 7955

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7708

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-iota 6484  df-fun 6534  df-fv 6540  df-1st 7957

This theorem is referenced by:  vafval  29719