Theorem df1st2 6187

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 6125	. . . . 5 ⊢ 1st :V–onto→V
2		fofn 5412	. . . . 5 ⊢ (1st :V–onto→V → 1st Fn V)
3		dffn5im 5532	. . . . 5 ⊢ (1st Fn V → 1st = (𝑤 ∈ V ↦ (1st ‘𝑤)))
4	1, 2, 3	mp2b 8	. . . 4 ⊢ 1st = (𝑤 ∈ V ↦ (1st ‘𝑤))
5		mptv 4079	. . . 4 ⊢ (𝑤 ∈ V ↦ (1st ‘𝑤)) = {⟨𝑤, 𝑧⟩ ∣ 𝑧 = (1st ‘𝑤)}
6	4, 5	eqtri 2186	. . 3 ⊢ 1st = {⟨𝑤, 𝑧⟩ ∣ 𝑧 = (1st ‘𝑤)}
7	6	reseq1i 4880	. 2 ⊢ (1st ↾ (V × V)) = ({⟨𝑤, 𝑧⟩ ∣ 𝑧 = (1st ‘𝑤)} ↾ (V × V))
8		resopab 4928	. 2 ⊢ ({⟨𝑤, 𝑧⟩ ∣ 𝑧 = (1st ‘𝑤)} ↾ (V × V)) = {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (V × V) ∧ 𝑧 = (1st ‘𝑤))}
9		vex 2729	. . . . 5 ⊢ 𝑥 ∈ V
10		vex 2729	. . . . 5 ⊢ 𝑦 ∈ V
11	9, 10	op1std 6116	. . . 4 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → (1st ‘𝑤) = 𝑥)
12	11	eqeq2d 2177	. . 3 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → (𝑧 = (1st ‘𝑤) ↔ 𝑧 = 𝑥))
13	12	dfoprab3 6159	. 2 ⊢ {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (V × V) ∧ 𝑧 = (1st ‘𝑤))} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥}
14	7, 8, 13	3eqtrri 2191	1 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥} = (1st ↾ (V × V))

Syntax hints:   ∧ wa 103   = wceq 1343   ∈ wcel 2136  Vcvv 2726  ⟨cop 3579  {copab 4042   ↦ cmpt 4043   × cxp 4602   ↾ cres 4606   Fn wfn 5183  –onto→wfo 5186  ‘cfv 5188  {coprab 5843  1^st c1st 6106

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-sbc 2952  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-fo 5194  df-fv 5196  df-oprab 5846  df-1st 6108  df-2nd 6109

This theorem is referenced by: (None)