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| Mirrors > Home > MPE Home > Th. List > opabssxpd | Structured version Visualization version GIF version | ||
| Description: An ordered-pair class abstraction is a subset of a Cartesian product. Formerly part of proof for opabex2 8082. (Contributed by AV, 26-Nov-2021.) |
| Ref | Expression |
|---|---|
| opabssxpd.x | ⊢ ((𝜑 ∧ 𝜓) → 𝑥 ∈ 𝐴) |
| opabssxpd.y | ⊢ ((𝜑 ∧ 𝜓) → 𝑦 ∈ 𝐵) |
| Ref | Expression |
|---|---|
| opabssxpd | ⊢ (𝜑 → {〈𝑥, 𝑦〉 ∣ 𝜓} ⊆ (𝐴 × 𝐵)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-opab 5206 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ 𝜓} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜓)} | |
| 2 | simprl 771 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜓)) → 𝑧 = 〈𝑥, 𝑦〉) | |
| 3 | opabssxpd.x | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝜓) → 𝑥 ∈ 𝐴) | |
| 4 | opabssxpd.y | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝜓) → 𝑦 ∈ 𝐵) | |
| 5 | 3, 4 | opelxpd 5724 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → 〈𝑥, 𝑦〉 ∈ (𝐴 × 𝐵)) |
| 6 | 5 | adantrl 716 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜓)) → 〈𝑥, 𝑦〉 ∈ (𝐴 × 𝐵)) |
| 7 | 2, 6 | eqeltrd 2841 | . . . . 5 ⊢ ((𝜑 ∧ (𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜓)) → 𝑧 ∈ (𝐴 × 𝐵)) |
| 8 | 7 | ex 412 | . . . 4 ⊢ (𝜑 → ((𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜓) → 𝑧 ∈ (𝐴 × 𝐵))) |
| 9 | 8 | exlimdvv 1934 | . . 3 ⊢ (𝜑 → (∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜓) → 𝑧 ∈ (𝐴 × 𝐵))) |
| 10 | 9 | abssdv 4068 | . 2 ⊢ (𝜑 → {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜓)} ⊆ (𝐴 × 𝐵)) |
| 11 | 1, 10 | eqsstrid 4022 | 1 ⊢ (𝜑 → {〈𝑥, 𝑦〉 ∣ 𝜓} ⊆ (𝐴 × 𝐵)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∃wex 1779 ∈ wcel 2108 {cab 2714 ⊆ wss 3951 〈cop 4632 {copab 5205 × cxp 5683 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-opab 5206 df-xp 5691 |
| This theorem is referenced by: opabex2 8082 erlval 33262 uspgropssxp 48060 |
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