Theorem 1st0 7992

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 7988	. 2 ⊢ (1st ‘∅) = ∪ dom {∅}
2		dmsn0 6211	. . 3 ⊢ dom {∅} = ∅
3	2	unieqi 4888	. 2 ⊢ ∪ dom {∅} = ∪ ∅
4		uni0 4905	. 2 ⊢ ∪ ∅ = ∅
5	1, 3, 4	3eqtri 2796	1 ⊢ (1st ‘∅) = ∅

Syntax hints:   = wceq 1567  ∅c0 4294  {csn 4594  ∪ cuni 4876  dom cdm 5662  ‘cfv 6537  1^st c1st 7984

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-iota 6493  df-fun 6539  df-fv 6545  df-1st 7986

This theorem is referenced by:  vafval  30896  fucofvalne  49988  reldmprcof1  50044  prcof1  50051