Theorem df2nd2 6169

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 6108	. . . . 5 ⊢ 2nd :V–onto→V
2		fofn 5396	. . . . 5 ⊢ (2nd :V–onto→V → 2nd Fn V)
3		dffn5im 5516	. . . . 5 ⊢ (2nd Fn V → 2nd = (𝑤 ∈ V ↦ (2nd ‘𝑤)))
4	1, 2, 3	mp2b 8	. . . 4 ⊢ 2nd = (𝑤 ∈ V ↦ (2nd ‘𝑤))
5		mptv 4063	. . . 4 ⊢ (𝑤 ∈ V ↦ (2nd ‘𝑤)) = {⟨𝑤, 𝑧⟩ ∣ 𝑧 = (2nd ‘𝑤)}
6	4, 5	eqtri 2178	. . 3 ⊢ 2nd = {⟨𝑤, 𝑧⟩ ∣ 𝑧 = (2nd ‘𝑤)}
7	6	reseq1i 4864	. 2 ⊢ (2nd ↾ (V × V)) = ({⟨𝑤, 𝑧⟩ ∣ 𝑧 = (2nd ‘𝑤)} ↾ (V × V))
8		resopab 4912	. 2 ⊢ ({⟨𝑤, 𝑧⟩ ∣ 𝑧 = (2nd ‘𝑤)} ↾ (V × V)) = {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (V × V) ∧ 𝑧 = (2nd ‘𝑤))}
9		vex 2715	. . . . 5 ⊢ 𝑥 ∈ V
10		vex 2715	. . . . 5 ⊢ 𝑦 ∈ V
11	9, 10	op2ndd 6099	. . . 4 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → (2nd ‘𝑤) = 𝑦)
12	11	eqeq2d 2169	. . 3 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → (𝑧 = (2nd ‘𝑤) ↔ 𝑧 = 𝑦))
13	12	dfoprab3 6141	. 2 ⊢ {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (V × V) ∧ 𝑧 = (2nd ‘𝑤))} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦}
14	7, 8, 13	3eqtrri 2183	1 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦} = (2nd ↾ (V × V))

Syntax hints:   ∧ wa 103   = wceq 1335   ∈ wcel 2128  Vcvv 2712  ⟨cop 3564  {copab 4026   ↦ cmpt 4027   × cxp 4586   ↾ cres 4590   Fn wfn 5167  –onto→wfo 5170  ‘cfv 5172  {coprab 5827  2^nd c2nd 6089

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4084  ax-pow 4137  ax-pr 4171  ax-un 4395

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-sbc 2938  df-un 3106  df-in 3108  df-ss 3115  df-pw 3546  df-sn 3567  df-pr 3568  df-op 3570  df-uni 3775  df-br 3968  df-opab 4028  df-mpt 4029  df-id 4255  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-iota 5137  df-fun 5174  df-fn 5175  df-f 5176  df-fo 5178  df-fv 5180  df-oprab 5830  df-1st 6090  df-2nd 6091

This theorem is referenced by: (None)