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Mirrors > Home > MPE Home > Th. List > opabbrex | Structured version Visualization version GIF version |
Description: A collection of ordered pairs with an extension of a binary relation is a set. (Contributed by Alexander van der Vekens, 1-Nov-2017.) (Revised by BJ/AV, 20-Jun-2019.) (Proof shortened by OpenAI, 25-Mar-2020.) |
Ref | Expression |
---|---|
opabbrex | ⊢ ((∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝜑) ∧ {〈𝑥, 𝑦〉 ∣ 𝜑} ∈ 𝑉) → {〈𝑥, 𝑦〉 ∣ (𝑥𝑅𝑦 ∧ 𝜓)} ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 488 | . 2 ⊢ ((∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝜑) ∧ {〈𝑥, 𝑦〉 ∣ 𝜑} ∈ 𝑉) → {〈𝑥, 𝑦〉 ∣ 𝜑} ∈ 𝑉) | |
2 | pm3.41 496 | . . . . 5 ⊢ ((𝑥𝑅𝑦 → 𝜑) → ((𝑥𝑅𝑦 ∧ 𝜓) → 𝜑)) | |
3 | 2 | 2alimi 1814 | . . . 4 ⊢ (∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝜑) → ∀𝑥∀𝑦((𝑥𝑅𝑦 ∧ 𝜓) → 𝜑)) |
4 | 3 | adantr 484 | . . 3 ⊢ ((∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝜑) ∧ {〈𝑥, 𝑦〉 ∣ 𝜑} ∈ 𝑉) → ∀𝑥∀𝑦((𝑥𝑅𝑦 ∧ 𝜓) → 𝜑)) |
5 | ssopab2 5398 | . . 3 ⊢ (∀𝑥∀𝑦((𝑥𝑅𝑦 ∧ 𝜓) → 𝜑) → {〈𝑥, 𝑦〉 ∣ (𝑥𝑅𝑦 ∧ 𝜓)} ⊆ {〈𝑥, 𝑦〉 ∣ 𝜑}) | |
6 | 4, 5 | syl 17 | . 2 ⊢ ((∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝜑) ∧ {〈𝑥, 𝑦〉 ∣ 𝜑} ∈ 𝑉) → {〈𝑥, 𝑦〉 ∣ (𝑥𝑅𝑦 ∧ 𝜓)} ⊆ {〈𝑥, 𝑦〉 ∣ 𝜑}) |
7 | 1, 6 | ssexd 5192 | 1 ⊢ ((∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝜑) ∧ {〈𝑥, 𝑦〉 ∣ 𝜑} ∈ 𝑉) → {〈𝑥, 𝑦〉 ∣ (𝑥𝑅𝑦 ∧ 𝜓)} ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∀wal 1536 ∈ wcel 2111 Vcvv 3441 ⊆ wss 3881 class class class wbr 5030 {copab 5092 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-ext 2770 ax-sep 5167 |
This theorem depends on definitions: df-bi 210 df-an 400 df-tru 1541 df-ex 1782 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-rab 3115 df-v 3443 df-in 3888 df-ss 3898 df-opab 5093 |
This theorem is referenced by: opabresex2d 7187 fvmptopab 7188 sprmpod 7873 wlkRes 27439 opabresex0d 43841 |
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