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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 3998 | . . 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 2228 | . 2 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → {𝑧 ∣ ∃𝑓∃𝑝(𝑧 = 〈𝑓, 𝑝〉 ∧ 𝜃)} ∈ V) |
4 | df-opab 3998 | . . 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 277 | . . . . . 6 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → ((𝑓(𝑉𝑊𝐸)𝑝 ∧ 𝜓) → 𝜃)) |
7 | 6 | anim2d 335 | . . . . 5 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → ((𝑧 = 〈𝑓, 𝑝〉 ∧ (𝑓(𝑉𝑊𝐸)𝑝 ∧ 𝜓)) → (𝑧 = 〈𝑓, 𝑝〉 ∧ 𝜃))) |
8 | 7 | 2eximdv 1855 | . . . 4 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (∃𝑓∃𝑝(𝑧 = 〈𝑓, 𝑝〉 ∧ (𝑓(𝑉𝑊𝐸)𝑝 ∧ 𝜓)) → ∃𝑓∃𝑝(𝑧 = 〈𝑓, 𝑝〉 ∧ 𝜃))) |
9 | 8 | ss2abdv 3175 | . . 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 3148 | . 2 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → {〈𝑓, 𝑝〉 ∣ (𝑓(𝑉𝑊𝐸)𝑝 ∧ 𝜓)} ⊆ {𝑧 ∣ ∃𝑓∃𝑝(𝑧 = 〈𝑓, 𝑝〉 ∧ 𝜃)}) |
11 | 3, 10 | ssexd 4076 | 1 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → {〈𝑓, 𝑝〉 ∣ (𝑓(𝑉𝑊𝐸)𝑝 ∧ 𝜓)} ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1332 ∃wex 1469 ∈ wcel 1481 {cab 2126 Vcvv 2689 〈cop 3535 class class class wbr 3937 {copab 3996 (class class class)co 5782 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 |
This theorem depends on definitions: df-bi 116 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-v 2691 df-in 3082 df-ss 3089 df-opab 3998 |
This theorem is referenced by: (None) |
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