Theorem 1st0 7700

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 7696	. 2 ⊢ (1st ‘∅) = ∪ dom {∅}
2		dmsn0 6039	. . 3 ⊢ dom {∅} = ∅
3	2	unieqi 4812	. 2 ⊢ ∪ dom {∅} = ∪ ∅
4		uni0 4829	. 2 ⊢ ∪ ∅ = ∅
5	1, 3, 4	3eqtri 2786	1 ⊢ (1st ‘∅) = ∅

Syntax hints:   = wceq 1539  ∅c0 4226  {csn 4523  ∪ cuni 4799  dom cdm 5525  ‘cfv 6336  1^st c1st 7692

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-sep 5170  ax-nul 5177  ax-pr 5299  ax-un 7460

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-ral 3076  df-rex 3077  df-rab 3080  df-v 3412  df-sbc 3698  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-nul 4227  df-if 4422  df-sn 4524  df-pr 4526  df-op 4530  df-uni 4800  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5431  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-iota 6295  df-fun 6338  df-fv 6344  df-1st 7694

This theorem is referenced by:  vafval  28486