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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 7757. (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 5131 | . 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 5595 | . . . . . . 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 2915 | . . . . 5 ⊢ ((𝜑 ∧ (𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜓)) → 𝑧 ∈ (𝐴 × 𝐵)) |
8 | 7 | ex 415 | . . . 4 ⊢ (𝜑 → ((𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜓) → 𝑧 ∈ (𝐴 × 𝐵))) |
9 | 8 | exlimdvv 1935 | . . 3 ⊢ (𝜑 → (∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜓) → 𝑧 ∈ (𝐴 × 𝐵))) |
10 | 9 | abssdv 4047 | . 2 ⊢ (𝜑 → {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜓)} ⊆ (𝐴 × 𝐵)) |
11 | 1, 10 | eqsstrid 4017 | 1 ⊢ (𝜑 → {〈𝑥, 𝑦〉 ∣ 𝜓} ⊆ (𝐴 × 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∃wex 1780 ∈ wcel 2114 {cab 2801 ⊆ wss 3938 〈cop 4575 {copab 5130 × cxp 5555 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-opab 5131 df-xp 5563 |
This theorem is referenced by: opabex2 7757 uspgropssxp 44026 |
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