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Theorem opabssxp 4696
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 4273 . 2 {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝜑)} ⊆ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝐵)}
3 df-xp 4628 . 2 (𝐴 × 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝐵)}
42, 3sseqtrri 3190 1 {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝜑)} ⊆ (𝐴 × 𝐵)
Colors of variables: wff set class
Syntax hints:  wa 104  wcel 2148  wss 3129  {copab 4060   × cxp 4620
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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-in 3135  df-ss 3142  df-opab 4062  df-xp 4628
This theorem is referenced by:  brab2ga  4697  dmoprabss  5950  ecopovsym  6624  ecopovtrn  6625  ecopover  6626  ecopovsymg  6627  ecopovtrng  6628  ecopoverg  6629  enqex  7337  ltrelnq  7342  enq0ex  7416  ltrelpr  7482  enrex  7714  ltrelsr  7715  ltrelre  7810  ltrelxr  7995  dvdszrcl  11770  lmfval  13325
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