Theorem 1st0 6306

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		0ex 4216	. . 3 ⊢ ∅ ∈ V
2		1stvalg 6304	. . 3 ⊢ (∅ ∈ V → (1st ‘∅) = ∪ dom {∅})
3	1, 2	ax-mp 5	. 2 ⊢ (1st ‘∅) = ∪ dom {∅}
4		dmsn0 5204	. . 3 ⊢ dom {∅} = ∅
5	4	unieqi 3903	. 2 ⊢ ∪ dom {∅} = ∪ ∅
6		uni0 3920	. 2 ⊢ ∪ ∅ = ∅
7	3, 5, 6	3eqtri 2256	1 ⊢ (1st ‘∅) = ∅

Syntax hints:   = wceq 1397   ∈ wcel 2202  Vcvv 2802  ∅c0 3494  {csn 3669  ∪ cuni 3893  dom cdm 4725  ‘cfv 5326  1^st c1st 6300

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-v 2804  df-sbc 3032  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-iota 5286  df-fun 5328  df-fv 5334  df-1st 6302

This theorem is referenced by:  0npr  7702