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Theorem df1st2 8123
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 8034 . . . . . 6 1st :V–onto→V
2 fofn 6822 . . . . . 6 (1st :V–onto→V → 1st Fn V)
31, 2ax-mp 5 . . . . 5 1st Fn V
4 dffn5 6967 . . . . 5 (1st Fn V ↔ 1st = (𝑤 ∈ V ↦ (1st𝑤)))
53, 4mpbi 230 . . . 4 1st = (𝑤 ∈ V ↦ (1st𝑤))
6 mptv 5258 . . . 4 (𝑤 ∈ V ↦ (1st𝑤)) = {⟨𝑤, 𝑧⟩ ∣ 𝑧 = (1st𝑤)}
75, 6eqtri 2765 . . 3 1st = {⟨𝑤, 𝑧⟩ ∣ 𝑧 = (1st𝑤)}
87reseq1i 5993 . 2 (1st ↾ (V × V)) = ({⟨𝑤, 𝑧⟩ ∣ 𝑧 = (1st𝑤)} ↾ (V × V))
9 resopab 6052 . 2 ({⟨𝑤, 𝑧⟩ ∣ 𝑧 = (1st𝑤)} ↾ (V × V)) = {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (V × V) ∧ 𝑧 = (1st𝑤))}
10 vex 3484 . . . . 5 𝑥 ∈ V
11 vex 3484 . . . . 5 𝑦 ∈ V
1210, 11op1std 8024 . . . 4 (𝑤 = ⟨𝑥, 𝑦⟩ → (1st𝑤) = 𝑥)
1312eqeq2d 2748 . . 3 (𝑤 = ⟨𝑥, 𝑦⟩ → (𝑧 = (1st𝑤) ↔ 𝑧 = 𝑥))
1413dfoprab3 8079 . 2 {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (V × V) ∧ 𝑧 = (1st𝑤))} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥}
158, 9, 143eqtrri 2770 1 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥} = (1st ↾ (V × V))
Colors of variables: wff setvar class
Syntax hints:  wa 395   = wceq 1540  wcel 2108  Vcvv 3480  cop 4632  {copab 5205  cmpt 5225   × cxp 5683  cres 5687   Fn wfn 6556  ontowfo 6559  cfv 6561  {coprab 7432  1st c1st 8012
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fo 6567  df-fv 6569  df-oprab 7435  df-1st 8014  df-2nd 8015
This theorem is referenced by:  df1stres  32713
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