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Theorem opabssxp 5164
Description: An abstraction relation is a subset of a related Cartesian product. (Contributed by NM, 16-Jul-1995.)
Assertion
Ref Expression
opabssxp {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝜑)} ⊆ (𝐴 × 𝐵)
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem opabssxp
StepHypRef Expression
1 simpl 473 . . 3 (((𝑥𝐴𝑦𝐵) ∧ 𝜑) → (𝑥𝐴𝑦𝐵))
21ssopab2i 4973 . 2 {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝜑)} ⊆ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝐵)}
3 df-xp 5090 . 2 (𝐴 × 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝐵)}
42, 3sseqtr4i 3623 1 {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝜑)} ⊆ (𝐴 × 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wa 384  wcel 1987  wss 3560  {copab 4682   × cxp 5082
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-in 3567  df-ss 3574  df-opab 4684  df-xp 5090
This theorem is referenced by:  brab2ga  5165  dmoprabss  6707  ecopovsym  7809  ecopovtrn  7810  ecopover  7811  ecopoverOLD  7812  enqex  9704  lterpq  9752  ltrelpr  9780  enrex  9848  ltrelsr  9849  ltrelre  9915  ltrelxr  10059  rlimpm  14181  dvdszrcl  14931  prdsle  16062  prdsless  16063  sectfval  16351  sectss  16352  ltbval  19411  opsrle  19415  lmfval  20976  isphtpc  22733  bcthlem1  23061  bcthlem5  23065  lgsquadlem1  25039  lgsquadlem2  25040  lgsquadlem3  25041  ishlg  25431  perpln1  25539  perpln2  25540  isperp  25541  iscgra  25635  isinag  25663  isleag  25667  inftmrel  29561  isinftm  29562  metidval  29757  metidss  29758  faeval  30132  filnetlem2  32069  areacirc  33176  lcvfbr  33826  cmtfvalN  34016  cvrfval  34074  dicssdvh  35994  pellexlem3  36914  pellexlem4  36915  pellexlem5  36916  pellex  36918  rfovcnvf1od  37819  fsovrfovd  37824
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