ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  opabbrex GIF version

Theorem opabbrex 5577
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.)
Hypotheses
Ref Expression
opabbrex.1 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑓(𝑉𝑊𝐸)𝑝𝜃))
opabbrex.2 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → {⟨𝑓, 𝑝⟩ ∣ 𝜃} ∈ V)
Assertion
Ref Expression
opabbrex ((𝑉 ∈ V ∧ 𝐸 ∈ V) → {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑉𝑊𝐸)𝑝𝜓)} ∈ V)
Distinct variable groups:   𝑓,𝐸,𝑝   𝑓,𝑉,𝑝
Allowed substitution hints:   𝜓(𝑓,𝑝)   𝜃(𝑓,𝑝)   𝑊(𝑓,𝑝)

Proof of Theorem opabbrex
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 df-opab 3847 . . 3 {⟨𝑓, 𝑝⟩ ∣ 𝜃} = {𝑧 ∣ ∃𝑓𝑝(𝑧 = ⟨𝑓, 𝑝⟩ ∧ 𝜃)}
2 opabbrex.2 . . 3 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → {⟨𝑓, 𝑝⟩ ∣ 𝜃} ∈ V)
31, 2syl5eqelr 2141 . 2 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → {𝑧 ∣ ∃𝑓𝑝(𝑧 = ⟨𝑓, 𝑝⟩ ∧ 𝜃)} ∈ V)
4 df-opab 3847 . . 3 {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑉𝑊𝐸)𝑝𝜓)} = {𝑧 ∣ ∃𝑓𝑝(𝑧 = ⟨𝑓, 𝑝⟩ ∧ (𝑓(𝑉𝑊𝐸)𝑝𝜓))}
5 opabbrex.1 . . . . . . 7 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑓(𝑉𝑊𝐸)𝑝𝜃))
65adantrd 268 . . . . . 6 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → ((𝑓(𝑉𝑊𝐸)𝑝𝜓) → 𝜃))
76anim2d 324 . . . . 5 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → ((𝑧 = ⟨𝑓, 𝑝⟩ ∧ (𝑓(𝑉𝑊𝐸)𝑝𝜓)) → (𝑧 = ⟨𝑓, 𝑝⟩ ∧ 𝜃)))
872eximdv 1778 . . . 4 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (∃𝑓𝑝(𝑧 = ⟨𝑓, 𝑝⟩ ∧ (𝑓(𝑉𝑊𝐸)𝑝𝜓)) → ∃𝑓𝑝(𝑧 = ⟨𝑓, 𝑝⟩ ∧ 𝜃)))
98ss2abdv 3041 . . 3 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → {𝑧 ∣ ∃𝑓𝑝(𝑧 = ⟨𝑓, 𝑝⟩ ∧ (𝑓(𝑉𝑊𝐸)𝑝𝜓))} ⊆ {𝑧 ∣ ∃𝑓𝑝(𝑧 = ⟨𝑓, 𝑝⟩ ∧ 𝜃)})
104, 9syl5eqss 3017 . 2 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑉𝑊𝐸)𝑝𝜓)} ⊆ {𝑧 ∣ ∃𝑓𝑝(𝑧 = ⟨𝑓, 𝑝⟩ ∧ 𝜃)})
113, 10ssexd 3925 1 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑉𝑊𝐸)𝑝𝜓)} ∈ V)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101   = wceq 1259  wex 1397  wcel 1409  {cab 2042  Vcvv 2574  cop 3406   class class class wbr 3792  {copab 3845  (class class class)co 5540
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-in 2952  df-ss 2959  df-opab 3847
This theorem is referenced by:  sprmpt2  5888
  Copyright terms: Public domain W3C validator