![]() |
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 4298 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑)} ⊆ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)} |
3 | df-xp 4653 | . 2 ⊢ (𝐴 × 𝐵) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)} | |
4 | 2, 3 | sseqtrri 3205 | 1 ⊢ {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 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 |