Theorem df2nd2 8078

Description: An alternate possible definition of the 2^nd function. (Contributed by NM, 10-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)

Assertion

Ref	Expression
df2nd2	⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦} = (2nd ↾ (V × V))

Distinct variable group:   𝑥,𝑦,𝑧

Proof of Theorem df2nd2

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fo2nd 7989	. . . . . 6 ⊢ 2nd :V–onto→V
2		fofn 6774	. . . . . 6 ⊢ (2nd :V–onto→V → 2nd Fn V)
3	1, 2	ax-mp 5	. . . . 5 ⊢ 2nd Fn V
4		dffn5 6919	. . . . 5 ⊢ (2nd Fn V ↔ 2nd = (𝑤 ∈ V ↦ (2nd ‘𝑤)))
5	3, 4	mpbi 230	. . . 4 ⊢ 2nd = (𝑤 ∈ V ↦ (2nd ‘𝑤))
6		mptv 5213	. . . 4 ⊢ (𝑤 ∈ V ↦ (2nd ‘𝑤)) = {⟨𝑤, 𝑧⟩ ∣ 𝑧 = (2nd ‘𝑤)}
7	5, 6	eqtri 2752	. . 3 ⊢ 2nd = {⟨𝑤, 𝑧⟩ ∣ 𝑧 = (2nd ‘𝑤)}
8	7	reseq1i 5946	. 2 ⊢ (2nd ↾ (V × V)) = ({⟨𝑤, 𝑧⟩ ∣ 𝑧 = (2nd ‘𝑤)} ↾ (V × V))
9		resopab 6005	. 2 ⊢ ({⟨𝑤, 𝑧⟩ ∣ 𝑧 = (2nd ‘𝑤)} ↾ (V × V)) = {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (V × V) ∧ 𝑧 = (2nd ‘𝑤))}
10		vex 3451	. . . . 5 ⊢ 𝑥 ∈ V
11		vex 3451	. . . . 5 ⊢ 𝑦 ∈ V
12	10, 11	op2ndd 7979	. . . 4 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → (2nd ‘𝑤) = 𝑦)
13	12	eqeq2d 2740	. . 3 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → (𝑧 = (2nd ‘𝑤) ↔ 𝑧 = 𝑦))
14	13	dfoprab3 8033	. 2 ⊢ {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (V × V) ∧ 𝑧 = (2nd ‘𝑤))} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦}
15	8, 9, 14	3eqtrri 2757	1 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦} = (2nd ↾ (V × V))

Syntax hints:   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3447  ⟨cop 4595  {copab 5169   ↦ cmpt 5188   × cxp 5636   ↾ cres 5640   Fn wfn 6506  –onto→wfo 6509  ‘cfv 6511  {coprab 7388  2^nd c2nd 7967

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-sbc 3754  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-fo 6517  df-fv 6519  df-oprab 7391  df-1st 7968  df-2nd 7969

This theorem is referenced by:  df2ndres  32628