Theorem 1st0 7922

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 7918	. 2 ⊢ (1st ‘∅) = ∪ dom {∅}
2		dmsn0 6151	. . 3 ⊢ dom {∅} = ∅
3	2	unieqi 4866	. 2 ⊢ ∪ dom {∅} = ∪ ∅
4		uni0 4882	. 2 ⊢ ∪ ∅ = ∅
5	1, 3, 4	3eqtri 2758	1 ⊢ (1st ‘∅) = ∅

Syntax hints:   = wceq 1541  ∅c0 4278  {csn 4571  ∪ cuni 4854  dom cdm 5611  ‘cfv 6476  1^st c1st 7914

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 5229  ax-nul 5239  ax-pr 5365  ax-un 7663

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 4279  df-if 4471  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5506  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-rn 5622  df-iota 6432  df-fun 6478  df-fv 6484  df-1st 7916

This theorem is referenced by:  vafval  30575  fucofvalne  49357  reldmprcof1  49413  prcof1  49420