Theorem 1st0 7937

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 7933	. 2 ⊢ (1st ‘∅) = ∪ dom {∅}
2		dmsn0 6160	. . 3 ⊢ dom {∅} = ∅
3	2	unieqi 4850	. 2 ⊢ ∪ dom {∅} = ∪ ∅
4		uni0 4866	. 2 ⊢ ∪ ∅ = ∅
5	1, 3, 4	3eqtri 2766	1 ⊢ (1st ‘∅) = ∅

Syntax hints:   = wceq 1547  ∅c0 4261  {csn 4555  ∪ cuni 4838  dom cdm 5618  ‘cfv 6485  1^st c1st 7929

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-iota 6441  df-fun 6487  df-fv 6493  df-1st 7931

This theorem is referenced by:  vafval  30692  fucofvalne  49815  reldmprcof1  49871  prcof1  49878