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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 7816. (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 5106 | . 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 5578 | . . . . . . 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 2834 | . . . . 5 ⊢ ((𝜑 ∧ (𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜓)) → 𝑧 ∈ (𝐴 × 𝐵)) |
8 | 7 | ex 416 | . . . 4 ⊢ (𝜑 → ((𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜓) → 𝑧 ∈ (𝐴 × 𝐵))) |
9 | 8 | exlimdvv 1942 | . . 3 ⊢ (𝜑 → (∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜓) → 𝑧 ∈ (𝐴 × 𝐵))) |
10 | 9 | abssdv 3972 | . 2 ⊢ (𝜑 → {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜓)} ⊆ (𝐴 × 𝐵)) |
11 | 1, 10 | eqsstrid 3939 | 1 ⊢ (𝜑 → {〈𝑥, 𝑦〉 ∣ 𝜓} ⊆ (𝐴 × 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∃wex 1787 ∈ wcel 2110 {cab 2712 ⊆ wss 3857 〈cop 4537 {copab 5105 × cxp 5538 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-sep 5181 ax-nul 5188 ax-pr 5311 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ral 3059 df-rex 3060 df-rab 3063 df-v 3403 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-nul 4228 df-if 4430 df-sn 4532 df-pr 4534 df-op 4538 df-opab 5106 df-xp 5546 |
This theorem is referenced by: opabex2 7816 uspgropssxp 44933 |
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