Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  opabssxp GIF version

Theorem opabssxp 4657
 Description: An abstraction relation is a subset of a related cross 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 108 . . 3 (((𝑥𝐴𝑦𝐵) ∧ 𝜑) → (𝑥𝐴𝑦𝐵))
21ssopab2i 4236 . 2 {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝜑)} ⊆ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝐵)}
3 df-xp 4589 . 2 (𝐴 × 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝐵)}
42, 3sseqtrri 3163 1 {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝜑)} ⊆ (𝐴 × 𝐵)
 Colors of variables: wff set class Syntax hints:   ∧ wa 103   ∈ wcel 2128   ⊆ wss 3102  {copab 4024   × cxp 4581 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139 This theorem depends on definitions:  df-bi 116  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-in 3108  df-ss 3115  df-opab 4026  df-xp 4589 This theorem is referenced by:  brab2ga  4658  dmoprabss  5897  ecopovsym  6569  ecopovtrn  6570  ecopover  6571  ecopovsymg  6572  ecopovtrng  6573  ecopoverg  6574  enqex  7263  ltrelnq  7268  enq0ex  7342  ltrelpr  7408  enrex  7640  ltrelsr  7641  ltrelre  7736  ltrelxr  7921  dvdszrcl  11670  lmfval  12552
 Copyright terms: Public domain W3C validator