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Mirrors > Home > MPE Home > Th. List > Mathboxes > opabresex2d | Structured version Visualization version GIF version |
Description: Restrictions of a collection of ordered pairs of related elements are sets. (Contributed by Alexander van der Vekens, 1-Nov-2017.) (Revised by AV, 15-Jan-2021.) |
Ref | Expression |
---|---|
opabresex2d.1 | ⊢ ((𝜑 ∧ 𝑥(𝑊‘𝐺)𝑦) → 𝜓) |
opabresex2d.2 | ⊢ (𝜑 → {〈𝑥, 𝑦〉 ∣ 𝜓} ∈ 𝑉) |
Ref | Expression |
---|---|
opabresex2d | ⊢ (𝜑 → {〈𝑥, 𝑦〉 ∣ (𝑥(𝑊‘𝐺)𝑦 ∧ 𝜃)} ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opabresex2d.1 | . . . 4 ⊢ ((𝜑 ∧ 𝑥(𝑊‘𝐺)𝑦) → 𝜓) | |
2 | 1 | ex 449 | . . 3 ⊢ (𝜑 → (𝑥(𝑊‘𝐺)𝑦 → 𝜓)) |
3 | 2 | alrimivv 1843 | . 2 ⊢ (𝜑 → ∀𝑥∀𝑦(𝑥(𝑊‘𝐺)𝑦 → 𝜓)) |
4 | opabresex2d.2 | . 2 ⊢ (𝜑 → {〈𝑥, 𝑦〉 ∣ 𝜓} ∈ 𝑉) | |
5 | opabbrex 6593 | . 2 ⊢ ((∀𝑥∀𝑦(𝑥(𝑊‘𝐺)𝑦 → 𝜓) ∧ {〈𝑥, 𝑦〉 ∣ 𝜓} ∈ 𝑉) → {〈𝑥, 𝑦〉 ∣ (𝑥(𝑊‘𝐺)𝑦 ∧ 𝜃)} ∈ V) | |
6 | 3, 4, 5 | syl2anc 691 | 1 ⊢ (𝜑 → {〈𝑥, 𝑦〉 ∣ (𝑥(𝑊‘𝐺)𝑦 ∧ 𝜃)} ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∀wal 1473 ∈ wcel 1977 Vcvv 3173 class class class wbr 4583 {copab 4642 ‘cfv 5804 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-v 3175 df-in 3547 df-ss 3554 df-opab 4644 |
This theorem is referenced by: mptmpt2opabbrd 40335 |
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