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| Mirrors > Home > ILE Home > Th. List > opabbrex | 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.) |
| Ref | Expression |
|---|---|
| opabbrex.1 | ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑓(𝑉𝑊𝐸)𝑝 → 𝜃)) |
| opabbrex.2 | ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → {〈𝑓, 𝑝〉 ∣ 𝜃} ∈ V) |
| Ref | Expression |
|---|---|
| opabbrex | ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → {〈𝑓, 𝑝〉 ∣ (𝑓(𝑉𝑊𝐸)𝑝 ∧ 𝜓)} ∈ V) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-opab 4106 | . . 3 ⊢ {〈𝑓, 𝑝〉 ∣ 𝜃} = {𝑧 ∣ ∃𝑓∃𝑝(𝑧 = 〈𝑓, 𝑝〉 ∧ 𝜃)} | |
| 2 | opabbrex.2 | . . 3 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → {〈𝑓, 𝑝〉 ∣ 𝜃} ∈ V) | |
| 3 | 1, 2 | eqeltrrid 2293 | . 2 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → {𝑧 ∣ ∃𝑓∃𝑝(𝑧 = 〈𝑓, 𝑝〉 ∧ 𝜃)} ∈ V) |
| 4 | df-opab 4106 | . . 3 ⊢ {〈𝑓, 𝑝〉 ∣ (𝑓(𝑉𝑊𝐸)𝑝 ∧ 𝜓)} = {𝑧 ∣ ∃𝑓∃𝑝(𝑧 = 〈𝑓, 𝑝〉 ∧ (𝑓(𝑉𝑊𝐸)𝑝 ∧ 𝜓))} | |
| 5 | opabbrex.1 | . . . . . . 7 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑓(𝑉𝑊𝐸)𝑝 → 𝜃)) | |
| 6 | 5 | adantrd 279 | . . . . . 6 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → ((𝑓(𝑉𝑊𝐸)𝑝 ∧ 𝜓) → 𝜃)) |
| 7 | 6 | anim2d 337 | . . . . 5 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → ((𝑧 = 〈𝑓, 𝑝〉 ∧ (𝑓(𝑉𝑊𝐸)𝑝 ∧ 𝜓)) → (𝑧 = 〈𝑓, 𝑝〉 ∧ 𝜃))) |
| 8 | 7 | 2eximdv 1905 | . . . 4 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (∃𝑓∃𝑝(𝑧 = 〈𝑓, 𝑝〉 ∧ (𝑓(𝑉𝑊𝐸)𝑝 ∧ 𝜓)) → ∃𝑓∃𝑝(𝑧 = 〈𝑓, 𝑝〉 ∧ 𝜃))) |
| 9 | 8 | ss2abdv 3266 | . . 3 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → {𝑧 ∣ ∃𝑓∃𝑝(𝑧 = 〈𝑓, 𝑝〉 ∧ (𝑓(𝑉𝑊𝐸)𝑝 ∧ 𝜓))} ⊆ {𝑧 ∣ ∃𝑓∃𝑝(𝑧 = 〈𝑓, 𝑝〉 ∧ 𝜃)}) |
| 10 | 4, 9 | eqsstrid 3239 | . 2 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → {〈𝑓, 𝑝〉 ∣ (𝑓(𝑉𝑊𝐸)𝑝 ∧ 𝜓)} ⊆ {𝑧 ∣ ∃𝑓∃𝑝(𝑧 = 〈𝑓, 𝑝〉 ∧ 𝜃)}) |
| 11 | 3, 10 | ssexd 4184 | 1 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → {〈𝑓, 𝑝〉 ∣ (𝑓(𝑉𝑊𝐸)𝑝 ∧ 𝜓)} ∈ V) |
| Colors of variables: wff set class |
| Syntax hints: → wi 4 ∧ wa 104 = wceq 1373 ∃wex 1515 ∈ wcel 2176 {cab 2191 Vcvv 2772 〈cop 3636 class class class wbr 4044 {copab 4104 (class class class)co 5944 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 711 ax-5 1470 ax-7 1471 ax-gen 1472 ax-ie1 1516 ax-ie2 1517 ax-8 1527 ax-10 1528 ax-11 1529 ax-i12 1530 ax-bndl 1532 ax-4 1533 ax-17 1549 ax-i9 1553 ax-ial 1557 ax-i5r 1558 ax-ext 2187 ax-sep 4162 |
| This theorem depends on definitions: df-bi 117 df-tru 1376 df-nf 1484 df-sb 1786 df-clab 2192 df-cleq 2198 df-clel 2201 df-nfc 2337 df-v 2774 df-in 3172 df-ss 3179 df-opab 4106 |
| This theorem is referenced by: (None) |
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