Theorem df2nd2 7777

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 7692	. . . . . 6 ⊢ 2nd :V–onto→V
2		fofn 6567	. . . . . 6 ⊢ (2nd :V–onto→V → 2nd Fn V)
3	1, 2	ax-mp 5	. . . . 5 ⊢ 2nd Fn V
4		dffn5 6699	. . . . 5 ⊢ (2nd Fn V ↔ 2nd = (𝑤 ∈ V ↦ (2nd ‘𝑤)))
5	3, 4	mpbi 233	. . . 4 ⊢ 2nd = (𝑤 ∈ V ↦ (2nd ‘𝑤))
6		mptv 5135	. . . 4 ⊢ (𝑤 ∈ V ↦ (2nd ‘𝑤)) = {⟨𝑤, 𝑧⟩ ∣ 𝑧 = (2nd ‘𝑤)}
7	5, 6	eqtri 2821	. . 3 ⊢ 2nd = {⟨𝑤, 𝑧⟩ ∣ 𝑧 = (2nd ‘𝑤)}
8	7	reseq1i 5814	. 2 ⊢ (2nd ↾ (V × V)) = ({⟨𝑤, 𝑧⟩ ∣ 𝑧 = (2nd ‘𝑤)} ↾ (V × V))
9		resopab 5869	. 2 ⊢ ({⟨𝑤, 𝑧⟩ ∣ 𝑧 = (2nd ‘𝑤)} ↾ (V × V)) = {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (V × V) ∧ 𝑧 = (2nd ‘𝑤))}
10		vex 3444	. . . . 5 ⊢ 𝑥 ∈ V
11		vex 3444	. . . . 5 ⊢ 𝑦 ∈ V
12	10, 11	op2ndd 7682	. . . 4 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → (2nd ‘𝑤) = 𝑦)
13	12	eqeq2d 2809	. . 3 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → (𝑧 = (2nd ‘𝑤) ↔ 𝑧 = 𝑦))
14	13	dfoprab3 7734	. 2 ⊢ {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (V × V) ∧ 𝑧 = (2nd ‘𝑤))} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦}
15	8, 9, 14	3eqtrri 2826	1 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦} = (2nd ↾ (V × V))

Syntax hints:   ∧ wa 399   = wceq 1538   ∈ wcel 2111  Vcvv 3441  ⟨cop 4531  {copab 5092   ↦ cmpt 5110   × cxp 5517   ↾ cres 5521   Fn wfn 6319  –onto→wfo 6322  ‘cfv 6324  {coprab 7136  2^nd c2nd 7670

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-fo 6330  df-fv 6332  df-oprab 7139  df-1st 7671  df-2nd 7672

This theorem is referenced by:  df2ndres  30464