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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 6919 | . 2 ⊢ (𝑊‘𝐺) ∈ V | |
| 2 | elopabran 5567 | . . 3 ⊢ (𝑧 ∈ {〈𝑥, 𝑦〉 ∣ (𝑥(𝑊‘𝐺)𝑦 ∧ 𝜃)} → 𝑧 ∈ (𝑊‘𝐺)) | |
| 3 | 2 | ssriv 3987 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ (𝑥(𝑊‘𝐺)𝑦 ∧ 𝜃)} ⊆ (𝑊‘𝐺) |
| 4 | 1, 3 | ssexi 5322 | 1 ⊢ {〈𝑥, 𝑦〉 ∣ (𝑥(𝑊‘𝐺)𝑦 ∧ 𝜃)} ∈ V |
| Colors of variables: wff setvar class |
| Syntax hints: ∧ wa 395 ∈ wcel 2108 Vcvv 3480 class class class wbr 5143 {copab 5205 ‘cfv 6561 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-sn 4627 df-pr 4629 df-uni 4908 df-br 5144 df-opab 5206 df-iota 6514 df-fv 6569 |
| This theorem is referenced by: fvmptopab 7487 mptmpoopabbrd 8105 mptmpoopabbrdOLD 8106 |
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