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Mirrors > Home > MPE Home > Th. List > opabresex2 | 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.) Add disjoint variable conditions betweem 𝑊, 𝐺 and 𝑥, 𝑦 to remove hypotheses. (Revised by SN, 13-Dec-2024.) |
Ref | Expression |
---|---|
opabresex2 | ⊢ {〈𝑥, 𝑦〉 ∣ (𝑥(𝑊‘𝐺)𝑦 ∧ 𝜃)} ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvex 6853 | . 2 ⊢ (𝑊‘𝐺) ∈ V | |
2 | elopabran 5518 | . . 3 ⊢ (𝑧 ∈ {〈𝑥, 𝑦〉 ∣ (𝑥(𝑊‘𝐺)𝑦 ∧ 𝜃)} → 𝑧 ∈ (𝑊‘𝐺)) | |
3 | 2 | ssriv 3947 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ (𝑥(𝑊‘𝐺)𝑦 ∧ 𝜃)} ⊆ (𝑊‘𝐺) |
4 | 1, 3 | ssexi 5278 | 1 ⊢ {〈𝑥, 𝑦〉 ∣ (𝑥(𝑊‘𝐺)𝑦 ∧ 𝜃)} ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 397 ∈ wcel 2107 Vcvv 3444 class class class wbr 5104 {copab 5166 ‘cfv 6494 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2709 ax-sep 5255 ax-nul 5262 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2716 df-cleq 2730 df-clel 2816 df-ne 2943 df-rab 3407 df-v 3446 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-nul 4282 df-sn 4586 df-pr 4588 df-uni 4865 df-br 5105 df-opab 5167 df-iota 6446 df-fv 6502 |
This theorem is referenced by: fvmptopab 7406 mptmpoopabbrd 8006 |
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