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Theorem df2ndres 31926
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 8085 . . . 4 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦} = (2nd ↾ (V × V))
21reseq1i 5978 . . 3 ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦} ↾ (𝐴 × 𝐵)) = ((2nd ↾ (V × V)) ↾ (𝐴 × 𝐵))
3 resoprab 7526 . . 3 ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦} ↾ (𝐴 × 𝐵)) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝑦)}
4 resres 5995 . . . 4 ((2nd ↾ (V × V)) ↾ (𝐴 × 𝐵)) = (2nd ↾ ((V × V) ∩ (𝐴 × 𝐵)))
5 incom 4202 . . . . . 6 ((𝐴 × 𝐵) ∩ (V × V)) = ((V × V) ∩ (𝐴 × 𝐵))
6 xpss 5693 . . . . . . 7 (𝐴 × 𝐵) ⊆ (V × V)
7 df-ss 3966 . . . . . . 7 ((𝐴 × 𝐵) ⊆ (V × V) ↔ ((𝐴 × 𝐵) ∩ (V × V)) = (𝐴 × 𝐵))
86, 7mpbi 229 . . . . . 6 ((𝐴 × 𝐵) ∩ (V × V)) = (𝐴 × 𝐵)
95, 8eqtr3i 2763 . . . . 5 ((V × V) ∩ (𝐴 × 𝐵)) = (𝐴 × 𝐵)
109reseq2i 5979 . . . 4 (2nd ↾ ((V × V) ∩ (𝐴 × 𝐵))) = (2nd ↾ (𝐴 × 𝐵))
114, 10eqtri 2761 . . 3 ((2nd ↾ (V × V)) ↾ (𝐴 × 𝐵)) = (2nd ↾ (𝐴 × 𝐵))
122, 3, 113eqtr3ri 2770 . 2 (2nd ↾ (𝐴 × 𝐵)) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝑦)}
13 df-mpo 7414 . 2 (𝑥𝐴, 𝑦𝐵𝑦) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝑦)}
1412, 13eqtr4i 2764 1 (2nd ↾ (𝐴 × 𝐵)) = (𝑥𝐴, 𝑦𝐵𝑦)
Colors of variables: wff setvar class
Syntax hints:  wa 397   = wceq 1542  wcel 2107  Vcvv 3475  cin 3948  wss 3949   × cxp 5675  cres 5679  {coprab 7410  cmpo 7411  2nd c2nd 7974
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fo 6550  df-fv 6552  df-oprab 7413  df-mpo 7414  df-1st 7975  df-2nd 7976
This theorem is referenced by:  cnre2csqima  32891
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