Theorem df2nd2 7788

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 7704	. . . . . 6 ⊢ 2nd :V–onto→V
2		fofn 6586	. . . . . 6 ⊢ (2nd :V–onto→V → 2nd Fn V)
3	1, 2	ax-mp 5	. . . . 5 ⊢ 2nd Fn V
4		dffn5 6718	. . . . 5 ⊢ (2nd Fn V ↔ 2nd = (𝑤 ∈ V ↦ (2nd ‘𝑤)))
5	3, 4	mpbi 232	. . . 4 ⊢ 2nd = (𝑤 ∈ V ↦ (2nd ‘𝑤))
6		mptv 5163	. . . 4 ⊢ (𝑤 ∈ V ↦ (2nd ‘𝑤)) = {⟨𝑤, 𝑧⟩ ∣ 𝑧 = (2nd ‘𝑤)}
7	5, 6	eqtri 2844	. . 3 ⊢ 2nd = {⟨𝑤, 𝑧⟩ ∣ 𝑧 = (2nd ‘𝑤)}
8	7	reseq1i 5843	. 2 ⊢ (2nd ↾ (V × V)) = ({⟨𝑤, 𝑧⟩ ∣ 𝑧 = (2nd ‘𝑤)} ↾ (V × V))
9		resopab 5896	. 2 ⊢ ({⟨𝑤, 𝑧⟩ ∣ 𝑧 = (2nd ‘𝑤)} ↾ (V × V)) = {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (V × V) ∧ 𝑧 = (2nd ‘𝑤))}
10		vex 3497	. . . . 5 ⊢ 𝑥 ∈ V
11		vex 3497	. . . . 5 ⊢ 𝑦 ∈ V
12	10, 11	op2ndd 7694	. . . 4 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → (2nd ‘𝑤) = 𝑦)
13	12	eqeq2d 2832	. . 3 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → (𝑧 = (2nd ‘𝑤) ↔ 𝑧 = 𝑦))
14	13	dfoprab3 7746	. 2 ⊢ {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (V × V) ∧ 𝑧 = (2nd ‘𝑤))} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦}
15	8, 9, 14	3eqtrri 2849	1 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦} = (2nd ↾ (V × V))

Syntax hints:   ∧ wa 398   = wceq 1533   ∈ wcel 2110  Vcvv 3494  ⟨cop 4566  {copab 5120   ↦ cmpt 5138   × cxp 5547   ↾ cres 5551   Fn wfn 6344  –onto→wfo 6347  ‘cfv 6349  {coprab 7151  2^nd c2nd 7682

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-br 5059  df-opab 5121  df-mpt 5139  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-fo 6355  df-fv 6357  df-oprab 7154  df-1st 7683  df-2nd 7684

This theorem is referenced by:  df2ndres  30434