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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 4064 | . . 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 2265 | . 2 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → {𝑧 ∣ ∃𝑓∃𝑝(𝑧 = ⟨𝑓, 𝑝⟩ ∧ 𝜃)} ∈ V) |
4 | df-opab 4064 | . . 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 1882 | . . . 4 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (∃𝑓∃𝑝(𝑧 = ⟨𝑓, 𝑝⟩ ∧ (𝑓(𝑉𝑊𝐸)𝑝 ∧ 𝜓)) → ∃𝑓∃𝑝(𝑧 = ⟨𝑓, 𝑝⟩ ∧ 𝜃))) |
9 | 8 | ss2abdv 3228 | . . 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 3201 | . 2 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑉𝑊𝐸)𝑝 ∧ 𝜓)} ⊆ {𝑧 ∣ ∃𝑓∃𝑝(𝑧 = ⟨𝑓, 𝑝⟩ ∧ 𝜃)}) |
11 | 3, 10 | ssexd 4142 | 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 1353 ∃wex 1492 ∈ wcel 2148 {cab 2163 Vcvv 2737 ⟨cop 3595 class class class wbr 4002 {copab 4062 (class class class)co 5872 |
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 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 ax-sep 4120 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-v 2739 df-in 3135 df-ss 3142 df-opab 4064 |
This theorem is referenced by: (None) |
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