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Theorem df2nd2 5841
 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 5785 . . . . 5 2nd :V–onto→V
2 fofn 5108 . . . . 5 (2nd :V–onto→V → 2nd Fn V)
3 dffn5im 5219 . . . . 5 (2nd Fn V → 2nd = (𝑤 ∈ V ↦ (2nd𝑤)))
41, 2, 3mp2b 8 . . . 4 2nd = (𝑤 ∈ V ↦ (2nd𝑤))
5 mptv 3853 . . . 4 (𝑤 ∈ V ↦ (2nd𝑤)) = {⟨𝑤, 𝑧⟩ ∣ 𝑧 = (2nd𝑤)}
64, 5eqtri 2060 . . 3 2nd = {⟨𝑤, 𝑧⟩ ∣ 𝑧 = (2nd𝑤)}
76reseq1i 4608 . 2 (2nd ↾ (V × V)) = ({⟨𝑤, 𝑧⟩ ∣ 𝑧 = (2nd𝑤)} ↾ (V × V))
8 resopab 4652 . 2 ({⟨𝑤, 𝑧⟩ ∣ 𝑧 = (2nd𝑤)} ↾ (V × V)) = {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (V × V) ∧ 𝑧 = (2nd𝑤))}
9 vex 2560 . . . . 5 𝑥 ∈ V
10 vex 2560 . . . . 5 𝑦 ∈ V
119, 10op2ndd 5776 . . . 4 (𝑤 = ⟨𝑥, 𝑦⟩ → (2nd𝑤) = 𝑦)
1211eqeq2d 2051 . . 3 (𝑤 = ⟨𝑥, 𝑦⟩ → (𝑧 = (2nd𝑤) ↔ 𝑧 = 𝑦))
1312dfoprab3 5817 . 2 {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (V × V) ∧ 𝑧 = (2nd𝑤))} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦}
147, 8, 133eqtrri 2065 1 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦} = (2nd ↾ (V × V))
 Colors of variables: wff set class Syntax hints:   ∧ wa 97   = wceq 1243   ∈ wcel 1393  Vcvv 2557  ⟨cop 3378  {copab 3817   ↦ cmpt 3818   × cxp 4343   ↾ cres 4347   Fn wfn 4897  –onto→wfo 4900  ‘cfv 4902  {coprab 5513  2nd c2nd 5766 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-fo 4908  df-fv 4910  df-oprab 5516  df-1st 5767  df-2nd 5768 This theorem is referenced by: (None)
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