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Theorem df2nd2 6085
Description: An alternate possible definition of the 2nd function. (Contributed by NM, 10-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)
Assertion
Ref Expression
df2nd2 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦} = (2nd ↾ (V × V))
Distinct variable group:   𝑥,𝑦,𝑧

Proof of Theorem df2nd2
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 fo2nd 6024 . . . . 5 2nd :V–onto→V
2 fofn 5317 . . . . 5 (2nd :V–onto→V → 2nd Fn V)
3 dffn5im 5435 . . . . 5 (2nd Fn V → 2nd = (𝑤 ∈ V ↦ (2nd𝑤)))
41, 2, 3mp2b 8 . . . 4 2nd = (𝑤 ∈ V ↦ (2nd𝑤))
5 mptv 3995 . . . 4 (𝑤 ∈ V ↦ (2nd𝑤)) = {⟨𝑤, 𝑧⟩ ∣ 𝑧 = (2nd𝑤)}
64, 5eqtri 2138 . . 3 2nd = {⟨𝑤, 𝑧⟩ ∣ 𝑧 = (2nd𝑤)}
76reseq1i 4785 . 2 (2nd ↾ (V × V)) = ({⟨𝑤, 𝑧⟩ ∣ 𝑧 = (2nd𝑤)} ↾ (V × V))
8 resopab 4833 . 2 ({⟨𝑤, 𝑧⟩ ∣ 𝑧 = (2nd𝑤)} ↾ (V × V)) = {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (V × V) ∧ 𝑧 = (2nd𝑤))}
9 vex 2663 . . . . 5 𝑥 ∈ V
10 vex 2663 . . . . 5 𝑦 ∈ V
119, 10op2ndd 6015 . . . 4 (𝑤 = ⟨𝑥, 𝑦⟩ → (2nd𝑤) = 𝑦)
1211eqeq2d 2129 . . 3 (𝑤 = ⟨𝑥, 𝑦⟩ → (𝑧 = (2nd𝑤) ↔ 𝑧 = 𝑦))
1312dfoprab3 6057 . 2 {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (V × V) ∧ 𝑧 = (2nd𝑤))} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦}
147, 8, 133eqtrri 2143 1 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦} = (2nd ↾ (V × V))
Colors of variables: wff set class
Syntax hints:  wa 103   = wceq 1316  wcel 1465  Vcvv 2660  cop 3500  {copab 3958  cmpt 3959   × cxp 4507  cres 4511   Fn wfn 5088  ontowfo 5091  cfv 5093  {coprab 5743  2nd c2nd 6005
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-sbc 2883  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-fo 5099  df-fv 5101  df-oprab 5746  df-1st 6006  df-2nd 6007
This theorem is referenced by: (None)
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