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

Theorem opabssxp 4721
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 109 . . 3 (((𝑥𝐴𝑦𝐵) ∧ 𝜑) → (𝑥𝐴𝑦𝐵))
21ssopab2i 4298 . 2 {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝜑)} ⊆ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝐵)}
3 df-xp 4653 . 2 (𝐴 × 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝐵)}
42, 3sseqtrri 3205 1 {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝜑)} ⊆ (𝐴 × 𝐵)
Colors of variables: wff set class
Syntax hints:  wa 104  wcel 2160  wss 3144  {copab 4081   × cxp 4645
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-in 3150  df-ss 3157  df-opab 4083  df-xp 4653
This theorem is referenced by:  brab2ga  4722  dmoprabss  5982  ecopovsym  6661  ecopovtrn  6662  ecopover  6663  ecopovsymg  6664  ecopovtrng  6665  ecopoverg  6666  netap  7288  2omotaplemap  7291  2omotaplemst  7292  enqex  7394  ltrelnq  7399  enq0ex  7473  ltrelpr  7539  enrex  7771  ltrelsr  7772  ltrelre  7867  ltrelxr  8053  dvdszrcl  11840  prdsex  12785  releqgg  13184  eqgex  13185  aprval  13623  aprap  13627  lmfval  14177
  Copyright terms: Public domain W3C validator