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Theorem df1st2 7500
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 7421 . . . . . 6 1st :V–onto→V
2 fofn 6333 . . . . . 6 (1st :V–onto→V → 1st Fn V)
31, 2ax-mp 5 . . . . 5 1st Fn V
4 dffn5 6466 . . . . 5 (1st Fn V ↔ 1st = (𝑤 ∈ V ↦ (1st𝑤)))
53, 4mpbi 222 . . . 4 1st = (𝑤 ∈ V ↦ (1st𝑤))
6 mptv 4944 . . . 4 (𝑤 ∈ V ↦ (1st𝑤)) = {⟨𝑤, 𝑧⟩ ∣ 𝑧 = (1st𝑤)}
75, 6eqtri 2821 . . 3 1st = {⟨𝑤, 𝑧⟩ ∣ 𝑧 = (1st𝑤)}
87reseq1i 5596 . 2 (1st ↾ (V × V)) = ({⟨𝑤, 𝑧⟩ ∣ 𝑧 = (1st𝑤)} ↾ (V × V))
9 resopab 5658 . 2 ({⟨𝑤, 𝑧⟩ ∣ 𝑧 = (1st𝑤)} ↾ (V × V)) = {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (V × V) ∧ 𝑧 = (1st𝑤))}
10 vex 3388 . . . . 5 𝑥 ∈ V
11 vex 3388 . . . . 5 𝑦 ∈ V
1210, 11op1std 7411 . . . 4 (𝑤 = ⟨𝑥, 𝑦⟩ → (1st𝑤) = 𝑥)
1312eqeq2d 2809 . . 3 (𝑤 = ⟨𝑥, 𝑦⟩ → (𝑧 = (1st𝑤) ↔ 𝑧 = 𝑥))
1413dfoprab3 7459 . 2 {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (V × V) ∧ 𝑧 = (1st𝑤))} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥}
158, 9, 143eqtrri 2826 1 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥} = (1st ↾ (V × V))
Colors of variables: wff setvar class
Syntax hints:  wa 385   = wceq 1653  wcel 2157  Vcvv 3385  cop 4374  {copab 4905  cmpt 4922   × cxp 5310  cres 5314   Fn wfn 6096  ontowfo 6099  cfv 6101  {coprab 6879  1st c1st 7399
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-fo 6107  df-fv 6109  df-oprab 6882  df-1st 7401  df-2nd 7402
This theorem is referenced by:  df1stres  29999
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