Theorem 2nd0 5954

Description: The value of the second-member function at the empty set. (Contributed by NM, 23-Apr-2007.)

Assertion

Ref	Expression
2nd0	⊢ (2nd ‘∅) = ∅

Proof of Theorem 2nd0

Step	Hyp	Ref	Expression
1		0ex 3987	. . 3 ⊢ ∅ ∈ V
2		2ndvalg 5952	. . 3 ⊢ (∅ ∈ V → (2nd ‘∅) = ∪ ran {∅})
3	1, 2	ax-mp 7	. 2 ⊢ (2nd ‘∅) = ∪ ran {∅}
4		dmsn0 4932	. . . 4 ⊢ dom {∅} = ∅
5		dm0rn0 4684	. . . 4 ⊢ (dom {∅} = ∅ ↔ ran {∅} = ∅)
6	4, 5	mpbi 144	. . 3 ⊢ ran {∅} = ∅
7	6	unieqi 3685	. 2 ⊢ ∪ ran {∅} = ∪ ∅
8		uni0 3702	. 2 ⊢ ∪ ∅ = ∅
9	3, 7, 8	3eqtri 2119	1 ⊢ (2nd ‘∅) = ∅

Syntax hints:   = wceq 1296   ∈ wcel 1445  Vcvv 2633  ∅c0 3302  {csn 3466  ∪ cuni 3675  dom cdm 4467  ran crn 4468  ‘cfv 5049  2^nd c2nd 5948

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-nul 3986  ax-pow 4030  ax-pr 4060  ax-un 4284

This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-fal 1302  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ne 2263  df-ral 2375  df-rex 2376  df-v 2635  df-sbc 2855  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026  df-nul 3303  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-br 3868  df-opab 3922  df-mpt 3923  df-id 4144  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-rn 4478  df-iota 5014  df-fun 5051  df-fv 5057  df-2nd 5950

This theorem is referenced by: (None)