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Mirrors > Home > ILE Home > Th. List > resopab2 | GIF version |
Description: Restriction of a class abstraction of ordered pairs. (Contributed by NM, 24-Aug-2007.) |
Ref | Expression |
---|---|
resopab2 | ⊢ (𝐴 ⊆ 𝐵 → ({〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐵 ∧ 𝜑)} ↾ 𝐴) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | resopab 4756 | . 2 ⊢ ({〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐵 ∧ 𝜑)} ↾ 𝐴) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ (𝑥 ∈ 𝐵 ∧ 𝜑))} | |
2 | ssel 3019 | . . . . . 6 ⊢ (𝐴 ⊆ 𝐵 → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)) | |
3 | 2 | pm4.71d 385 | . . . . 5 ⊢ (𝐴 ⊆ 𝐵 → (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵))) |
4 | 3 | anbi1d 453 | . . . 4 ⊢ (𝐴 ⊆ 𝐵 → ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ∧ 𝜑))) |
5 | anass 393 | . . . 4 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ∧ 𝜑) ↔ (𝑥 ∈ 𝐴 ∧ (𝑥 ∈ 𝐵 ∧ 𝜑))) | |
6 | 4, 5 | syl6rbb 195 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → ((𝑥 ∈ 𝐴 ∧ (𝑥 ∈ 𝐵 ∧ 𝜑)) ↔ (𝑥 ∈ 𝐴 ∧ 𝜑))) |
7 | 6 | opabbidv 3904 | . 2 ⊢ (𝐴 ⊆ 𝐵 → {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ (𝑥 ∈ 𝐵 ∧ 𝜑))} = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}) |
8 | 1, 7 | syl5eq 2132 | 1 ⊢ (𝐴 ⊆ 𝐵 → ({〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐵 ∧ 𝜑)} ↾ 𝐴) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 = wceq 1289 ∈ wcel 1438 ⊆ wss 2999 {copab 3898 ↾ cres 4440 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-14 1450 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 ax-sep 3957 ax-pow 4009 ax-pr 4036 |
This theorem depends on definitions: df-bi 115 df-3an 926 df-tru 1292 df-nf 1395 df-sb 1693 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-ral 2364 df-rex 2365 df-v 2621 df-un 3003 df-in 3005 df-ss 3012 df-pw 3431 df-sn 3452 df-pr 3453 df-op 3455 df-opab 3900 df-xp 4444 df-rel 4445 df-res 4450 |
This theorem is referenced by: resmpt 4760 |
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