Theorem 1st0 7695

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 7691	. 2 ⊢ (1st ‘∅) = ∪ dom {∅}
2		dmsn0 6066	. . 3 ⊢ dom {∅} = ∅
3	2	unieqi 4851	. 2 ⊢ ∪ dom {∅} = ∪ ∅
4		uni0 4866	. 2 ⊢ ∪ ∅ = ∅
5	1, 3, 4	3eqtri 2848	1 ⊢ (1st ‘∅) = ∅

Syntax hints:   = wceq 1537  ∅c0 4291  {csn 4567  ∪ cuni 4838  dom cdm 5555  ‘cfv 6355  1^st c1st 7687

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  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 3773  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-br 5067  df-opab 5129  df-mpt 5147  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-iota 6314  df-fun 6357  df-fv 6363  df-1st 7689

This theorem is referenced by:  vafval  28380