Theorem 1st0 7694

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 7690	. 2 ⊢ (1st ‘∅) = ∪ dom {∅}
2		dmsn0 6065	. . 3 ⊢ dom {∅} = ∅
3	2	unieqi 4850	. 2 ⊢ ∪ dom {∅} = ∪ ∅
4		uni0 4865	. 2 ⊢ ∪ ∅ = ∅
5	1, 3, 4	3eqtri 2848	1 ⊢ (1st ‘∅) = ∅

Syntax hints:   = wceq 1533  ∅c0 4290  {csn 4566  ∪ cuni 4837  dom cdm 5554  ‘cfv 6354  1^st c1st 7686

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-un 7460

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4838  df-br 5066  df-opab 5128  df-mpt 5146  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-iota 6313  df-fun 6356  df-fv 6362  df-1st 7688

This theorem is referenced by:  vafval  28379