Theorem 1st0 7941

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 7937	. 2 ⊢ (1st ‘∅) = ∪ dom {∅}
2		dmsn0 6167	. . 3 ⊢ dom {∅} = ∅
3	2	unieqi 4863	. 2 ⊢ ∪ dom {∅} = ∪ ∅
4		uni0 4879	. 2 ⊢ ∪ ∅ = ∅
5	1, 3, 4	3eqtri 2764	1 ⊢ (1st ‘∅) = ∅

Syntax hints:   = wceq 1542  ∅c0 4274  {csn 4568  ∪ cuni 4851  dom cdm 5624  ‘cfv 6492  1^st c1st 7933

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-nul 5241  ax-pr 5370  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-iota 6448  df-fun 6494  df-fv 6500  df-1st 7935

This theorem is referenced by:  vafval  30689  fucofvalne  49812  reldmprcof1  49868  prcof1  49875