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Theorem df1st2 6082
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 6021 . . . . 5 1st :V–onto→V
2 fofn 5315 . . . . 5 (1st :V–onto→V → 1st Fn V)
3 dffn5im 5433 . . . . 5 (1st Fn V → 1st = (𝑤 ∈ V ↦ (1st𝑤)))
41, 2, 3mp2b 8 . . . 4 1st = (𝑤 ∈ V ↦ (1st𝑤))
5 mptv 3993 . . . 4 (𝑤 ∈ V ↦ (1st𝑤)) = {⟨𝑤, 𝑧⟩ ∣ 𝑧 = (1st𝑤)}
64, 5eqtri 2136 . . 3 1st = {⟨𝑤, 𝑧⟩ ∣ 𝑧 = (1st𝑤)}
76reseq1i 4783 . 2 (1st ↾ (V × V)) = ({⟨𝑤, 𝑧⟩ ∣ 𝑧 = (1st𝑤)} ↾ (V × V))
8 resopab 4831 . 2 ({⟨𝑤, 𝑧⟩ ∣ 𝑧 = (1st𝑤)} ↾ (V × V)) = {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (V × V) ∧ 𝑧 = (1st𝑤))}
9 vex 2661 . . . . 5 𝑥 ∈ V
10 vex 2661 . . . . 5 𝑦 ∈ V
119, 10op1std 6012 . . . 4 (𝑤 = ⟨𝑥, 𝑦⟩ → (1st𝑤) = 𝑥)
1211eqeq2d 2127 . . 3 (𝑤 = ⟨𝑥, 𝑦⟩ → (𝑧 = (1st𝑤) ↔ 𝑧 = 𝑥))
1312dfoprab3 6055 . 2 {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (V × V) ∧ 𝑧 = (1st𝑤))} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥}
147, 8, 133eqtrri 2141 1 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥} = (1st ↾ (V × V))
Colors of variables: wff set class
Syntax hints:  wa 103   = wceq 1314  wcel 1463  Vcvv 2658  cop 3498  {copab 3956  cmpt 3957   × cxp 4505  cres 4509   Fn wfn 5086  ontowfo 5089  cfv 5091  {coprab 5741  1st c1st 6002
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099  ax-un 4323
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-sbc 2881  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-br 3898  df-opab 3958  df-mpt 3959  df-id 4183  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-fo 5097  df-fv 5099  df-oprab 5744  df-1st 6004  df-2nd 6005
This theorem is referenced by: (None)
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