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Mirrors > Home > ILE Home > Th. List > opabssxp | GIF version |
Description: An abstraction relation is a subset of a related cross product. (Contributed by NM, 16-Jul-1995.) |
Ref | Expression |
---|---|
opabssxp | ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑)} ⊆ (𝐴 × 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 109 | . . 3 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑) → (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) | |
2 | 1 | ssopab2i 4277 | . 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑)} ⊆ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)} |
3 | df-xp 4632 | . 2 ⊢ (𝐴 × 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)} | |
4 | 2, 3 | sseqtrri 3190 | 1 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑)} ⊆ (𝐴 × 𝐵) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 104 ∈ wcel 2148 ⊆ wss 3129 {copab 4063 × cxp 4624 |
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 4065 df-xp 4632 |
This theorem is referenced by: brab2ga 4701 dmoprabss 5956 ecopovsym 6630 ecopovtrn 6631 ecopover 6632 ecopovsymg 6633 ecopovtrng 6634 ecopoverg 6635 netap 7252 2omotaplemap 7255 2omotaplemst 7256 enqex 7358 ltrelnq 7363 enq0ex 7437 ltrelpr 7503 enrex 7735 ltrelsr 7736 ltrelre 7831 ltrelxr 8017 dvdszrcl 11798 prdsex 12717 releqgg 13078 aprval 13370 aprap 13374 lmfval 13662 |
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