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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 7985. (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 5166 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ 𝜓} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜓)} | |
2 | simprl 769 | . . . . . 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 5669 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → 〈𝑥, 𝑦〉 ∈ (𝐴 × 𝐵)) |
6 | 5 | adantrl 714 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜓)) → 〈𝑥, 𝑦〉 ∈ (𝐴 × 𝐵)) |
7 | 2, 6 | eqeltrd 2838 | . . . . 5 ⊢ ((𝜑 ∧ (𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜓)) → 𝑧 ∈ (𝐴 × 𝐵)) |
8 | 7 | ex 413 | . . . 4 ⊢ (𝜑 → ((𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜓) → 𝑧 ∈ (𝐴 × 𝐵))) |
9 | 8 | exlimdvv 1937 | . . 3 ⊢ (𝜑 → (∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜓) → 𝑧 ∈ (𝐴 × 𝐵))) |
10 | 9 | abssdv 4023 | . 2 ⊢ (𝜑 → {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜓)} ⊆ (𝐴 × 𝐵)) |
11 | 1, 10 | eqsstrid 3990 | 1 ⊢ (𝜑 → {〈𝑥, 𝑦〉 ∣ 𝜓} ⊆ (𝐴 × 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∃wex 1781 ∈ wcel 2106 {cab 2713 ⊆ wss 3908 〈cop 4590 {copab 5165 × cxp 5629 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2707 ax-sep 5254 ax-nul 5261 ax-pr 5382 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2714 df-cleq 2728 df-clel 2814 df-ral 3063 df-rex 3072 df-rab 3406 df-v 3445 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-sn 4585 df-pr 4587 df-op 4591 df-opab 5166 df-xp 5637 |
This theorem is referenced by: opabex2 7985 uspgropssxp 45978 |
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