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Theorem df2nd2 7769
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 7685 . . . . . 6 2nd :V–onto→V
2 fofn 6565 . . . . . 6 (2nd :V–onto→V → 2nd Fn V)
31, 2ax-mp 5 . . . . 5 2nd Fn V
4 dffn5 6697 . . . . 5 (2nd Fn V ↔ 2nd = (𝑤 ∈ V ↦ (2nd𝑤)))
53, 4mpbi 233 . . . 4 2nd = (𝑤 ∈ V ↦ (2nd𝑤))
6 mptv 5144 . . . 4 (𝑤 ∈ V ↦ (2nd𝑤)) = {⟨𝑤, 𝑧⟩ ∣ 𝑧 = (2nd𝑤)}
75, 6eqtri 2844 . . 3 2nd = {⟨𝑤, 𝑧⟩ ∣ 𝑧 = (2nd𝑤)}
87reseq1i 5822 . 2 (2nd ↾ (V × V)) = ({⟨𝑤, 𝑧⟩ ∣ 𝑧 = (2nd𝑤)} ↾ (V × V))
9 resopab 5875 . 2 ({⟨𝑤, 𝑧⟩ ∣ 𝑧 = (2nd𝑤)} ↾ (V × V)) = {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (V × V) ∧ 𝑧 = (2nd𝑤))}
10 vex 3474 . . . . 5 𝑥 ∈ V
11 vex 3474 . . . . 5 𝑦 ∈ V
1210, 11op2ndd 7675 . . . 4 (𝑤 = ⟨𝑥, 𝑦⟩ → (2nd𝑤) = 𝑦)
1312eqeq2d 2832 . . 3 (𝑤 = ⟨𝑥, 𝑦⟩ → (𝑧 = (2nd𝑤) ↔ 𝑧 = 𝑦))
1413dfoprab3 7727 . 2 {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (V × V) ∧ 𝑧 = (2nd𝑤))} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦}
158, 9, 143eqtrri 2849 1 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦} = (2nd ↾ (V × V))
Colors of variables: wff setvar class
Syntax hints:  wa 399   = wceq 1538  wcel 2115  Vcvv 3471  cop 4546  {copab 5101  cmpt 5119   × cxp 5526  cres 5530   Fn wfn 6323  ontowfo 6326  cfv 6328  {coprab 7131  2nd c2nd 7663
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ral 3131  df-rex 3132  df-rab 3135  df-v 3473  df-sbc 3750  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-br 5040  df-opab 5102  df-mpt 5120  df-id 5433  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-fo 6334  df-fv 6336  df-oprab 7134  df-1st 7664  df-2nd 7665
This theorem is referenced by:  df2ndres  30426
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