Theorem df1st2 6355

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 6293	. . . . 5 ⊢ 1st :V–onto→V
2		fofn 5546	. . . . 5 ⊢ (1st :V–onto→V → 1st Fn V)
3		dffn5im 5672	. . . . 5 ⊢ (1st Fn V → 1st = (𝑤 ∈ V ↦ (1st ‘𝑤)))
4	1, 2, 3	mp2b 8	. . . 4 ⊢ 1st = (𝑤 ∈ V ↦ (1st ‘𝑤))
5		mptv 4180	. . . 4 ⊢ (𝑤 ∈ V ↦ (1st ‘𝑤)) = {⟨𝑤, 𝑧⟩ ∣ 𝑧 = (1st ‘𝑤)}
6	4, 5	eqtri 2250	. . 3 ⊢ 1st = {⟨𝑤, 𝑧⟩ ∣ 𝑧 = (1st ‘𝑤)}
7	6	reseq1i 4997	. 2 ⊢ (1st ↾ (V × V)) = ({⟨𝑤, 𝑧⟩ ∣ 𝑧 = (1st ‘𝑤)} ↾ (V × V))
8		resopab 5045	. 2 ⊢ ({⟨𝑤, 𝑧⟩ ∣ 𝑧 = (1st ‘𝑤)} ↾ (V × V)) = {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (V × V) ∧ 𝑧 = (1st ‘𝑤))}
9		vex 2802	. . . . 5 ⊢ 𝑥 ∈ V
10		vex 2802	. . . . 5 ⊢ 𝑦 ∈ V
11	9, 10	op1std 6284	. . . 4 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → (1st ‘𝑤) = 𝑥)
12	11	eqeq2d 2241	. . 3 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → (𝑧 = (1st ‘𝑤) ↔ 𝑧 = 𝑥))
13	12	dfoprab3 6327	. 2 ⊢ {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (V × V) ∧ 𝑧 = (1st ‘𝑤))} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥}
14	7, 8, 13	3eqtrri 2255	1 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥} = (1st ↾ (V × V))

Syntax hints:   ∧ wa 104   = wceq 1395   ∈ wcel 2200  Vcvv 2799  ⟨cop 3669  {copab 4143   ↦ cmpt 4144   × cxp 4714   ↾ cres 4718   Fn wfn 5309  –onto→wfo 5312  ‘cfv 5314  {coprab 5995  1^st c1st 6274

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-un 4521

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-sbc 3029  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4381  df-xp 4722  df-rel 4723  df-cnv 4724  df-co 4725  df-dm 4726  df-rn 4727  df-res 4728  df-iota 5274  df-fun 5316  df-fn 5317  df-f 5318  df-fo 5320  df-fv 5322  df-oprab 5998  df-1st 6276  df-2nd 6277

This theorem is referenced by: (None)