![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
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 4279 | . 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑)} ⊆ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)} |
3 | df-xp 4634 | . 2 ⊢ (𝐴 × 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)} | |
4 | 2, 3 | sseqtrri 3192 | 1 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑)} ⊆ (𝐴 × 𝐵) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 104 ∈ wcel 2148 ⊆ wss 3131 {copab 4065 × cxp 4626 |
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 3137 df-ss 3144 df-opab 4067 df-xp 4634 |
This theorem is referenced by: brab2ga 4703 dmoprabss 5959 ecopovsym 6633 ecopovtrn 6634 ecopover 6635 ecopovsymg 6636 ecopovtrng 6637 ecopoverg 6638 netap 7255 2omotaplemap 7258 2omotaplemst 7259 enqex 7361 ltrelnq 7366 enq0ex 7440 ltrelpr 7506 enrex 7738 ltrelsr 7739 ltrelre 7834 ltrelxr 8020 dvdszrcl 11801 prdsex 12723 releqgg 13085 aprval 13377 aprap 13381 lmfval 13731 |
Copyright terms: Public domain | W3C validator |