Theorem df2nd2 6199

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 6137	. . . . 5 ⊢ 2nd :V–onto→V
2		fofn 5422	. . . . 5 ⊢ (2nd :V–onto→V → 2nd Fn V)
3		dffn5im 5542	. . . . 5 ⊢ (2nd Fn V → 2nd = (𝑤 ∈ V ↦ (2nd ‘𝑤)))
4	1, 2, 3	mp2b 8	. . . 4 ⊢ 2nd = (𝑤 ∈ V ↦ (2nd ‘𝑤))
5		mptv 4086	. . . 4 ⊢ (𝑤 ∈ V ↦ (2nd ‘𝑤)) = {⟨𝑤, 𝑧⟩ ∣ 𝑧 = (2nd ‘𝑤)}
6	4, 5	eqtri 2191	. . 3 ⊢ 2nd = {⟨𝑤, 𝑧⟩ ∣ 𝑧 = (2nd ‘𝑤)}
7	6	reseq1i 4887	. 2 ⊢ (2nd ↾ (V × V)) = ({⟨𝑤, 𝑧⟩ ∣ 𝑧 = (2nd ‘𝑤)} ↾ (V × V))
8		resopab 4935	. 2 ⊢ ({⟨𝑤, 𝑧⟩ ∣ 𝑧 = (2nd ‘𝑤)} ↾ (V × V)) = {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (V × V) ∧ 𝑧 = (2nd ‘𝑤))}
9		vex 2733	. . . . 5 ⊢ 𝑥 ∈ V
10		vex 2733	. . . . 5 ⊢ 𝑦 ∈ V
11	9, 10	op2ndd 6128	. . . 4 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → (2nd ‘𝑤) = 𝑦)
12	11	eqeq2d 2182	. . 3 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → (𝑧 = (2nd ‘𝑤) ↔ 𝑧 = 𝑦))
13	12	dfoprab3 6170	. 2 ⊢ {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (V × V) ∧ 𝑧 = (2nd ‘𝑤))} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦}
14	7, 8, 13	3eqtrri 2196	1 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦} = (2nd ↾ (V × V))

Syntax hints:   ∧ wa 103   = wceq 1348   ∈ wcel 2141  Vcvv 2730  ⟨cop 3586  {copab 4049   ↦ cmpt 4050   × cxp 4609   ↾ cres 4613   Fn wfn 5193  –onto→wfo 5196  ‘cfv 5198  {coprab 5854  2^nd c2nd 6118

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-fo 5204  df-fv 5206  df-oprab 5857  df-1st 6119  df-2nd 6120

This theorem is referenced by: (None)