Theorem df1st2 6272

Description: An alternate possible definition of the 1^st function. (Contributed by NM, 14-Oct-2004.) (Revised by Mario Carneiro, 31-Aug-2015.)

Assertion

Ref	Expression
df1st2	⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥} = (1st ↾ (V × V))

Distinct variable group:   𝑥,𝑦,𝑧

Proof of Theorem df1st2

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fo1st 6210	. . . . 5 ⊢ 1st :V–onto→V
2		fofn 5478	. . . . 5 ⊢ (1st :V–onto→V → 1st Fn V)
3		dffn5im 5602	. . . . 5 ⊢ (1st Fn V → 1st = (𝑤 ∈ V ↦ (1st ‘𝑤)))
4	1, 2, 3	mp2b 8	. . . 4 ⊢ 1st = (𝑤 ∈ V ↦ (1st ‘𝑤))
5		mptv 4126	. . . 4 ⊢ (𝑤 ∈ V ↦ (1st ‘𝑤)) = {⟨𝑤, 𝑧⟩ ∣ 𝑧 = (1st ‘𝑤)}
6	4, 5	eqtri 2214	. . 3 ⊢ 1st = {⟨𝑤, 𝑧⟩ ∣ 𝑧 = (1st ‘𝑤)}
7	6	reseq1i 4938	. 2 ⊢ (1st ↾ (V × V)) = ({⟨𝑤, 𝑧⟩ ∣ 𝑧 = (1st ‘𝑤)} ↾ (V × V))
8		resopab 4986	. 2 ⊢ ({⟨𝑤, 𝑧⟩ ∣ 𝑧 = (1st ‘𝑤)} ↾ (V × V)) = {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (V × V) ∧ 𝑧 = (1st ‘𝑤))}
9		vex 2763	. . . . 5 ⊢ 𝑥 ∈ V
10		vex 2763	. . . . 5 ⊢ 𝑦 ∈ V
11	9, 10	op1std 6201	. . . 4 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → (1st ‘𝑤) = 𝑥)
12	11	eqeq2d 2205	. . 3 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → (𝑧 = (1st ‘𝑤) ↔ 𝑧 = 𝑥))
13	12	dfoprab3 6244	. 2 ⊢ {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (V × V) ∧ 𝑧 = (1st ‘𝑤))} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥}
14	7, 8, 13	3eqtrri 2219	1 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥} = (1st ↾ (V × V))

Syntax hints:   ∧ wa 104   = wceq 1364   ∈ wcel 2164  Vcvv 2760  ⟨cop 3621  {copab 4089   ↦ cmpt 4090   × cxp 4657   ↾ cres 4661   Fn wfn 5249  –onto→wfo 5252  ‘cfv 5254  {coprab 5919  1^st c1st 6191

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2986  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-fo 5260  df-fv 5262  df-oprab 5922  df-1st 6193  df-2nd 6194

This theorem is referenced by: (None)