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Theorem df2nd2 5783
 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 5727 . . . . 5 2nd :V–onto→V
2 fofn 5051 . . . . 5 (2nd :V–onto→V → 2nd Fn V)
3 dffn5im 5162 . . . . 5 (2nd Fn V → 2nd = (w V ↦ (2ndw)))
41, 2, 3mp2b 8 . . . 4 2nd = (w V ↦ (2ndw))
5 mptv 3844 . . . 4 (w V ↦ (2ndw)) = {⟨w, z⟩ ∣ z = (2ndw)}
64, 5eqtri 2057 . . 3 2nd = {⟨w, z⟩ ∣ z = (2ndw)}
76reseq1i 4551 . 2 (2nd ↾ (V × V)) = ({⟨w, z⟩ ∣ z = (2ndw)} ↾ (V × V))
8 resopab 4595 . 2 ({⟨w, z⟩ ∣ z = (2ndw)} ↾ (V × V)) = {⟨w, z⟩ ∣ (w (V × V) z = (2ndw))}
9 vex 2554 . . . . 5 x V
10 vex 2554 . . . . 5 y V
119, 10op2ndd 5718 . . . 4 (w = ⟨x, y⟩ → (2ndw) = y)
1211eqeq2d 2048 . . 3 (w = ⟨x, y⟩ → (z = (2ndw) ↔ z = y))
1312dfoprab3 5759 . 2 {⟨w, z⟩ ∣ (w (V × V) z = (2ndw))} = {⟨⟨x, y⟩, z⟩ ∣ z = y}
147, 8, 133eqtrri 2062 1 {⟨⟨x, y⟩, z⟩ ∣ z = y} = (2nd ↾ (V × V))
 Colors of variables: wff set class Syntax hints:   ∧ wa 97   = wceq 1242   ∈ wcel 1390  Vcvv 2551  ⟨cop 3370  {copab 3808   ↦ cmpt 3809   × cxp 4286   ↾ cres 4290   Fn wfn 4840  –onto→wfo 4843  ‘cfv 4845  {coprab 5456  2nd c2nd 5708 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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136 This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-fo 4851  df-fv 4853  df-oprab 5459  df-1st 5709  df-2nd 5710 This theorem is referenced by: (None)
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