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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 8042. (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 5204 | . 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜓} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)} | |
2 | simprl 768 | . . . . . 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 5708 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)) |
6 | 5 | adantrl 713 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)) → ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)) |
7 | 2, 6 | eqeltrd 2827 | . . . . 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 1929 | . . 3 ⊢ (𝜑 → (∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓) → 𝑧 ∈ (𝐴 × 𝐵))) |
10 | 9 | abssdv 4060 | . 2 ⊢ (𝜑 → {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)} ⊆ (𝐴 × 𝐵)) |
11 | 1, 10 | eqsstrid 4025 | 1 ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜓} ⊆ (𝐴 × 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∃wex 1773 ∈ wcel 2098 {cab 2703 ⊆ wss 3943 ⟨cop 4629 {copab 5203 × cxp 5667 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-opab 5204 df-xp 5675 |
This theorem is referenced by: opabex2 8042 uspgropssxp 47094 |
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