Theorem 1st0 8041

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 8037	. 2 ⊢ (1st ‘∅) = ∪ dom {∅}
2		dmsn0 6229	. . 3 ⊢ dom {∅} = ∅
3	2	unieqi 4884	. 2 ⊢ ∪ dom {∅} = ∪ ∅
4		uni0 4901	. 2 ⊢ ∪ ∅ = ∅
5	1, 3, 4	3eqtri 2790	1 ⊢ (1st ‘∅) = ∅

This proof depends on syntax axioms:   = wceq 1570  ∅c0 4281  {csn 4588  ∪ cuni 4872  dom cdm 5658  ‘cfv 6566  1^st c1st 8033

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pr 5398  ax-un 7783

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-iota 6520  df-fun 6568  df-fv 6574  df-1st 8035

This theorem is used by:  vafval  31262  fucofvalne  50778  reldmprcof1  50847  prcof1  50854