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Theorem df2nd2 7502
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 7423 . . . . . 6 2nd :V–onto→V
2 fofn 6334 . . . . . 6 (2nd :V–onto→V → 2nd Fn V)
31, 2ax-mp 5 . . . . 5 2nd Fn V
4 dffn5 6467 . . . . 5 (2nd Fn V ↔ 2nd = (𝑤 ∈ V ↦ (2nd𝑤)))
53, 4mpbi 222 . . . 4 2nd = (𝑤 ∈ V ↦ (2nd𝑤))
6 mptv 4945 . . . 4 (𝑤 ∈ V ↦ (2nd𝑤)) = {⟨𝑤, 𝑧⟩ ∣ 𝑧 = (2nd𝑤)}
75, 6eqtri 2822 . . 3 2nd = {⟨𝑤, 𝑧⟩ ∣ 𝑧 = (2nd𝑤)}
87reseq1i 5597 . 2 (2nd ↾ (V × V)) = ({⟨𝑤, 𝑧⟩ ∣ 𝑧 = (2nd𝑤)} ↾ (V × V))
9 resopab 5659 . 2 ({⟨𝑤, 𝑧⟩ ∣ 𝑧 = (2nd𝑤)} ↾ (V × V)) = {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (V × V) ∧ 𝑧 = (2nd𝑤))}
10 vex 3389 . . . . 5 𝑥 ∈ V
11 vex 3389 . . . . 5 𝑦 ∈ V
1210, 11op2ndd 7413 . . . 4 (𝑤 = ⟨𝑥, 𝑦⟩ → (2nd𝑤) = 𝑦)
1312eqeq2d 2810 . . 3 (𝑤 = ⟨𝑥, 𝑦⟩ → (𝑧 = (2nd𝑤) ↔ 𝑧 = 𝑦))
1413dfoprab3 7460 . 2 {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (V × V) ∧ 𝑧 = (2nd𝑤))} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦}
158, 9, 143eqtrri 2827 1 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦} = (2nd ↾ (V × V))
Colors of variables: wff setvar class
Syntax hints:  wa 385   = wceq 1653  wcel 2157  Vcvv 3386  cop 4375  {copab 4906  cmpt 4923   × cxp 5311  cres 5315   Fn wfn 6097  ontowfo 6100  cfv 6102  {coprab 6880  2nd c2nd 7401
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2378  ax-ext 2778  ax-sep 4976  ax-nul 4984  ax-pow 5036  ax-pr 5098  ax-un 7184
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2592  df-eu 2610  df-clab 2787  df-cleq 2793  df-clel 2796  df-nfc 2931  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3388  df-sbc 3635  df-dif 3773  df-un 3775  df-in 3777  df-ss 3784  df-nul 4117  df-if 4279  df-sn 4370  df-pr 4372  df-op 4376  df-uni 4630  df-br 4845  df-opab 4907  df-mpt 4924  df-id 5221  df-xp 5319  df-rel 5320  df-cnv 5321  df-co 5322  df-dm 5323  df-rn 5324  df-res 5325  df-iota 6065  df-fun 6104  df-fn 6105  df-f 6106  df-fo 6108  df-fv 6110  df-oprab 6883  df-1st 7402  df-2nd 7403
This theorem is referenced by:  df2ndres  29999
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