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Theorem df1st2 5867
 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 5811 . . . . 5 1st :V–onto→V
2 fofn 5135 . . . . 5 (1st :V–onto→V → 1st Fn V)
3 dffn5im 5246 . . . . 5 (1st Fn V → 1st = (𝑤 ∈ V ↦ (1st𝑤)))
41, 2, 3mp2b 8 . . . 4 1st = (𝑤 ∈ V ↦ (1st𝑤))
5 mptv 3880 . . . 4 (𝑤 ∈ V ↦ (1st𝑤)) = {⟨𝑤, 𝑧⟩ ∣ 𝑧 = (1st𝑤)}
64, 5eqtri 2076 . . 3 1st = {⟨𝑤, 𝑧⟩ ∣ 𝑧 = (1st𝑤)}
76reseq1i 4635 . 2 (1st ↾ (V × V)) = ({⟨𝑤, 𝑧⟩ ∣ 𝑧 = (1st𝑤)} ↾ (V × V))
8 resopab 4679 . 2 ({⟨𝑤, 𝑧⟩ ∣ 𝑧 = (1st𝑤)} ↾ (V × V)) = {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (V × V) ∧ 𝑧 = (1st𝑤))}
9 vex 2577 . . . . 5 𝑥 ∈ V
10 vex 2577 . . . . 5 𝑦 ∈ V
119, 10op1std 5802 . . . 4 (𝑤 = ⟨𝑥, 𝑦⟩ → (1st𝑤) = 𝑥)
1211eqeq2d 2067 . . 3 (𝑤 = ⟨𝑥, 𝑦⟩ → (𝑧 = (1st𝑤) ↔ 𝑧 = 𝑥))
1312dfoprab3 5844 . 2 {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (V × V) ∧ 𝑧 = (1st𝑤))} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥}
147, 8, 133eqtrri 2081 1 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥} = (1st ↾ (V × V))
 Colors of variables: wff set class Syntax hints:   ∧ wa 101   = wceq 1259   ∈ wcel 1409  Vcvv 2574  ⟨cop 3405  {copab 3844   ↦ cmpt 3845   × cxp 4370   ↾ cres 4374   Fn wfn 4924  –onto→wfo 4927  ‘cfv 4929  {coprab 5540  1st c1st 5792 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971  ax-un 4197 This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-sbc 2787  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-br 3792  df-opab 3846  df-mpt 3847  df-id 4057  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-rn 4383  df-res 4384  df-iota 4894  df-fun 4931  df-fn 4932  df-f 4933  df-fo 4935  df-fv 4937  df-oprab 5543  df-1st 5794  df-2nd 5795 This theorem is referenced by: (None)
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