Theorem 1st0 7974

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 7970	. 2 ⊢ (1st ‘∅) = ∪ dom {∅}
2		dmsn0 6182	. . 3 ⊢ dom {∅} = ∅
3	2	unieqi 4883	. 2 ⊢ ∪ dom {∅} = ∪ ∅
4		uni0 4899	. 2 ⊢ ∪ ∅ = ∅
5	1, 3, 4	3eqtri 2756	1 ⊢ (1st ‘∅) = ∅

Syntax hints:   = wceq 1540  ∅c0 4296  {csn 4589  ∪ cuni 4871  dom cdm 5638  ‘cfv 6511  1^st c1st 7966

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-iota 6464  df-fun 6513  df-fv 6519  df-1st 7968

This theorem is referenced by:  vafval  30532  fucofvalne  49314  reldmprcof1  49370  prcof1  49377