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Theorem df2ndres 30508
 Description: Definition for a restriction of the 2nd (second member of an ordered pair) function. (Contributed by Thierry Arnoux, 27-Sep-2017.)
Assertion
Ref Expression
df2ndres (2nd ↾ (𝐴 × 𝐵)) = (𝑥𝐴, 𝑦𝐵𝑦)
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦

Proof of Theorem df2ndres
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 df2nd2 7790 . . . 4 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦} = (2nd ↾ (V × V))
21reseq1i 5818 . . 3 ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦} ↾ (𝐴 × 𝐵)) = ((2nd ↾ (V × V)) ↾ (𝐴 × 𝐵))
3 resoprab 7259 . . 3 ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦} ↾ (𝐴 × 𝐵)) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝑦)}
4 resres 5835 . . . 4 ((2nd ↾ (V × V)) ↾ (𝐴 × 𝐵)) = (2nd ↾ ((V × V) ∩ (𝐴 × 𝐵)))
5 incom 4131 . . . . . 6 ((𝐴 × 𝐵) ∩ (V × V)) = ((V × V) ∩ (𝐴 × 𝐵))
6 xpss 5539 . . . . . . 7 (𝐴 × 𝐵) ⊆ (V × V)
7 df-ss 3900 . . . . . . 7 ((𝐴 × 𝐵) ⊆ (V × V) ↔ ((𝐴 × 𝐵) ∩ (V × V)) = (𝐴 × 𝐵))
86, 7mpbi 233 . . . . . 6 ((𝐴 × 𝐵) ∩ (V × V)) = (𝐴 × 𝐵)
95, 8eqtr3i 2823 . . . . 5 ((V × V) ∩ (𝐴 × 𝐵)) = (𝐴 × 𝐵)
109reseq2i 5819 . . . 4 (2nd ↾ ((V × V) ∩ (𝐴 × 𝐵))) = (2nd ↾ (𝐴 × 𝐵))
114, 10eqtri 2821 . . 3 ((2nd ↾ (V × V)) ↾ (𝐴 × 𝐵)) = (2nd ↾ (𝐴 × 𝐵))
122, 3, 113eqtr3ri 2830 . 2 (2nd ↾ (𝐴 × 𝐵)) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝑦)}
13 df-mpo 7150 . 2 (𝑥𝐴, 𝑦𝐵𝑦) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝑦)}
1412, 13eqtr4i 2824 1 (2nd ↾ (𝐴 × 𝐵)) = (𝑥𝐴, 𝑦𝐵𝑦)
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 399   = wceq 1538   ∈ wcel 2111  Vcvv 3442   ∩ cin 3882   ⊆ wss 3883   × cxp 5521   ↾ cres 5525  {coprab 7146   ∈ cmpo 7147  2nd c2nd 7683 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5171  ax-nul 5178  ax-pr 5299  ax-un 7454 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 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3444  df-sbc 3723  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4805  df-br 5035  df-opab 5097  df-mpt 5115  df-id 5429  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-iota 6291  df-fun 6334  df-fn 6335  df-f 6336  df-fo 6338  df-fv 6340  df-oprab 7149  df-mpo 7150  df-1st 7684  df-2nd 7685 This theorem is referenced by:  cnre2csqima  31330
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