Theorem 1st0 7320

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 7316	. 2 ⊢ (1st ‘∅) = ∪ dom {∅}
2		dmsn0 5743	. . 3 ⊢ dom {∅} = ∅
3	2	unieqi 4581	. 2 ⊢ ∪ dom {∅} = ∪ ∅
4		uni0 4599	. 2 ⊢ ∪ ∅ = ∅
5	1, 3, 4	3eqtri 2796	1 ⊢ (1st ‘∅) = ∅

Syntax hints:   = wceq 1630  ∅c0 4061  {csn 4314  ∪ cuni 4572  dom cdm 5249  ‘cfv 6031  1^st c1st 7312

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-iota 5994  df-fun 6033  df-fv 6039  df-1st 7314

This theorem is referenced by:  vafval  27792