Theorem 1st0 7933

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 7929	. 2 ⊢ (1st ‘∅) = ∪ dom {∅}
2		dmsn0 6162	. . 3 ⊢ dom {∅} = ∅
3	2	unieqi 4870	. 2 ⊢ ∪ dom {∅} = ∪ ∅
4		uni0 4886	. 2 ⊢ ∪ ∅ = ∅
5	1, 3, 4	3eqtri 2758	1 ⊢ (1st ‘∅) = ∅

Syntax hints:   = wceq 1541  ∅c0 4282  {csn 4575  ∪ cuni 4858  dom cdm 5619  ‘cfv 6487  1^st c1st 7925

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5236  ax-nul 5246  ax-pr 5372  ax-un 7674

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4283  df-if 4475  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-br 5094  df-opab 5156  df-mpt 5175  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-iota 6443  df-fun 6489  df-fv 6495  df-1st 7927

This theorem is referenced by:  vafval  30590  fucofvalne  49431  reldmprcof1  49487  prcof1  49494