Theorem df1st2 6116

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 6055	. . . . 5 ⊢ 1st :V–onto→V
2		fofn 5347	. . . . 5 ⊢ (1st :V–onto→V → 1st Fn V)
3		dffn5im 5467	. . . . 5 ⊢ (1st Fn V → 1st = (𝑤 ∈ V ↦ (1st ‘𝑤)))
4	1, 2, 3	mp2b 8	. . . 4 ⊢ 1st = (𝑤 ∈ V ↦ (1st ‘𝑤))
5		mptv 4025	. . . 4 ⊢ (𝑤 ∈ V ↦ (1st ‘𝑤)) = {⟨𝑤, 𝑧⟩ ∣ 𝑧 = (1st ‘𝑤)}
6	4, 5	eqtri 2160	. . 3 ⊢ 1st = {⟨𝑤, 𝑧⟩ ∣ 𝑧 = (1st ‘𝑤)}
7	6	reseq1i 4815	. 2 ⊢ (1st ↾ (V × V)) = ({⟨𝑤, 𝑧⟩ ∣ 𝑧 = (1st ‘𝑤)} ↾ (V × V))
8		resopab 4863	. 2 ⊢ ({⟨𝑤, 𝑧⟩ ∣ 𝑧 = (1st ‘𝑤)} ↾ (V × V)) = {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (V × V) ∧ 𝑧 = (1st ‘𝑤))}
9		vex 2689	. . . . 5 ⊢ 𝑥 ∈ V
10		vex 2689	. . . . 5 ⊢ 𝑦 ∈ V
11	9, 10	op1std 6046	. . . 4 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → (1st ‘𝑤) = 𝑥)
12	11	eqeq2d 2151	. . 3 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → (𝑧 = (1st ‘𝑤) ↔ 𝑧 = 𝑥))
13	12	dfoprab3 6089	. 2 ⊢ {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (V × V) ∧ 𝑧 = (1st ‘𝑤))} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥}
14	7, 8, 13	3eqtrri 2165	1 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥} = (1st ↾ (V × V))

Syntax hints:   ∧ wa 103   = wceq 1331   ∈ wcel 1480  Vcvv 2686  ⟨cop 3530  {copab 3988   ↦ cmpt 3989   × cxp 4537   ↾ cres 4541   Fn wfn 5118  –onto→wfo 5121  ‘cfv 5123  {coprab 5775  1^st c1st 6036

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-sbc 2910  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-fo 5129  df-fv 5131  df-oprab 5778  df-1st 6038  df-2nd 6039

This theorem is referenced by: (None)