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Theorem df2ndres 31797
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 8067 . . . 4 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦} = (2nd ↾ (V × V))
21reseq1i 5969 . . 3 ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦} ↾ (𝐴 × 𝐵)) = ((2nd ↾ (V × V)) ↾ (𝐴 × 𝐵))
3 resoprab 7510 . . 3 ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦} ↾ (𝐴 × 𝐵)) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝑦)}
4 resres 5986 . . . 4 ((2nd ↾ (V × V)) ↾ (𝐴 × 𝐵)) = (2nd ↾ ((V × V) ∩ (𝐴 × 𝐵)))
5 incom 4197 . . . . . 6 ((𝐴 × 𝐵) ∩ (V × V)) = ((V × V) ∩ (𝐴 × 𝐵))
6 xpss 5685 . . . . . . 7 (𝐴 × 𝐵) ⊆ (V × V)
7 df-ss 3961 . . . . . . 7 ((𝐴 × 𝐵) ⊆ (V × V) ↔ ((𝐴 × 𝐵) ∩ (V × V)) = (𝐴 × 𝐵))
86, 7mpbi 229 . . . . . 6 ((𝐴 × 𝐵) ∩ (V × V)) = (𝐴 × 𝐵)
95, 8eqtr3i 2761 . . . . 5 ((V × V) ∩ (𝐴 × 𝐵)) = (𝐴 × 𝐵)
109reseq2i 5970 . . . 4 (2nd ↾ ((V × V) ∩ (𝐴 × 𝐵))) = (2nd ↾ (𝐴 × 𝐵))
114, 10eqtri 2759 . . 3 ((2nd ↾ (V × V)) ↾ (𝐴 × 𝐵)) = (2nd ↾ (𝐴 × 𝐵))
122, 3, 113eqtr3ri 2768 . 2 (2nd ↾ (𝐴 × 𝐵)) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝑦)}
13 df-mpo 7398 . 2 (𝑥𝐴, 𝑦𝐵𝑦) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝑦)}
1412, 13eqtr4i 2762 1 (2nd ↾ (𝐴 × 𝐵)) = (𝑥𝐴, 𝑦𝐵𝑦)
Colors of variables: wff setvar class
Syntax hints:  wa 396   = wceq 1541  wcel 2106  Vcvv 3473  cin 3943  wss 3944   × cxp 5667  cres 5671  {coprab 7394  cmpo 7395  2nd c2nd 7956
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7708
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3774  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-iota 6484  df-fun 6534  df-fn 6535  df-f 6536  df-fo 6538  df-fv 6540  df-oprab 7397  df-mpo 7398  df-1st 7957  df-2nd 7958
This theorem is referenced by:  cnre2csqima  32722
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