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Mirrors > Home > MPE Home > Th. List > 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 403 | . . 3 ⊢ (𝜑 → (𝑥(𝑊‘𝐺)𝑦 → 𝜓)) |
3 | 2 | alrimivv 2027 | . 2 ⊢ (𝜑 → ∀𝑥∀𝑦(𝑥(𝑊‘𝐺)𝑦 → 𝜓)) |
4 | opabresex2d.2 | . 2 ⊢ (𝜑 → {〈𝑥, 𝑦〉 ∣ 𝜓} ∈ 𝑉) | |
5 | opabbrex 6960 | . 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 579 | 1 ⊢ (𝜑 → {〈𝑥, 𝑦〉 ∣ (𝑥(𝑊‘𝐺)𝑦 ∧ 𝜃)} ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 ∀wal 1654 ∈ wcel 2164 Vcvv 3414 class class class wbr 4875 {copab 4937 ‘cfv 6127 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5007 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-v 3416 df-in 3805 df-ss 3812 df-opab 4938 |
This theorem is referenced by: mptmpt2opabbrd 7516 |
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