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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 8081. (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 5211 | . 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 5728 | . . . . . . 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 2839 | . . . . 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 1932 | . . 3 ⊢ (𝜑 → (∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜓) → 𝑧 ∈ (𝐴 × 𝐵))) |
10 | 9 | abssdv 4078 | . 2 ⊢ (𝜑 → {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜓)} ⊆ (𝐴 × 𝐵)) |
11 | 1, 10 | eqsstrid 4044 | 1 ⊢ (𝜑 → {〈𝑥, 𝑦〉 ∣ 𝜓} ⊆ (𝐴 × 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∃wex 1776 ∈ wcel 2106 {cab 2712 ⊆ wss 3963 〈cop 4637 {copab 5210 × cxp 5687 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-opab 5211 df-xp 5695 |
This theorem is referenced by: opabex2 8081 erlval 33245 uspgropssxp 47988 |
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