ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  df1st2 GIF version

Theorem df1st2 5976
Description: An alternate possible definition of the 1st 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.
StepHypRef Expression
1 fo1st 5920 . . . . 5 1st :V–onto→V
2 fofn 5229 . . . . 5 (1st :V–onto→V → 1st Fn V)
3 dffn5im 5344 . . . . 5 (1st Fn V → 1st = (𝑤 ∈ V ↦ (1st𝑤)))
41, 2, 3mp2b 8 . . . 4 1st = (𝑤 ∈ V ↦ (1st𝑤))
5 mptv 3933 . . . 4 (𝑤 ∈ V ↦ (1st𝑤)) = {⟨𝑤, 𝑧⟩ ∣ 𝑧 = (1st𝑤)}
64, 5eqtri 2108 . . 3 1st = {⟨𝑤, 𝑧⟩ ∣ 𝑧 = (1st𝑤)}
76reseq1i 4705 . 2 (1st ↾ (V × V)) = ({⟨𝑤, 𝑧⟩ ∣ 𝑧 = (1st𝑤)} ↾ (V × V))
8 resopab 4751 . 2 ({⟨𝑤, 𝑧⟩ ∣ 𝑧 = (1st𝑤)} ↾ (V × V)) = {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (V × V) ∧ 𝑧 = (1st𝑤))}
9 vex 2622 . . . . 5 𝑥 ∈ V
10 vex 2622 . . . . 5 𝑦 ∈ V
119, 10op1std 5911 . . . 4 (𝑤 = ⟨𝑥, 𝑦⟩ → (1st𝑤) = 𝑥)
1211eqeq2d 2099 . . 3 (𝑤 = ⟨𝑥, 𝑦⟩ → (𝑧 = (1st𝑤) ↔ 𝑧 = 𝑥))
1312dfoprab3 5953 . 2 {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (V × V) ∧ 𝑧 = (1st𝑤))} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥}
147, 8, 133eqtrri 2113 1 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥} = (1st ↾ (V × V))
Colors of variables: wff set class
Syntax hints:  wa 102   = wceq 1289  wcel 1438  Vcvv 2619  cop 3447  {copab 3896  cmpt 3897   × cxp 4434  cres 4438   Fn wfn 5005  ontowfo 5008  cfv 5010  {coprab 5645  1st c1st 5901
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3955  ax-pow 4007  ax-pr 4034  ax-un 4258
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2841  df-un 3003  df-in 3005  df-ss 3012  df-pw 3429  df-sn 3450  df-pr 3451  df-op 3453  df-uni 3652  df-br 3844  df-opab 3898  df-mpt 3899  df-id 4118  df-xp 4442  df-rel 4443  df-cnv 4444  df-co 4445  df-dm 4446  df-rn 4447  df-res 4448  df-iota 4975  df-fun 5012  df-fn 5013  df-f 5014  df-fo 5016  df-fv 5018  df-oprab 5648  df-1st 5903  df-2nd 5904
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator