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Theorem opabbrex 5988
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 4105 . . 3 {⟨𝑓, 𝑝⟩ ∣ 𝜃} = {𝑧 ∣ ∃𝑓𝑝(𝑧 = ⟨𝑓, 𝑝⟩ ∧ 𝜃)}
2 opabbrex.2 . . 3 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → {⟨𝑓, 𝑝⟩ ∣ 𝜃} ∈ V)
31, 2eqeltrrid 2292 . 2 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → {𝑧 ∣ ∃𝑓𝑝(𝑧 = ⟨𝑓, 𝑝⟩ ∧ 𝜃)} ∈ V)
4 df-opab 4105 . . 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 279 . . . . . 6 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → ((𝑓(𝑉𝑊𝐸)𝑝𝜓) → 𝜃))
76anim2d 337 . . . . 5 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → ((𝑧 = ⟨𝑓, 𝑝⟩ ∧ (𝑓(𝑉𝑊𝐸)𝑝𝜓)) → (𝑧 = ⟨𝑓, 𝑝⟩ ∧ 𝜃)))
872eximdv 1904 . . . 4 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (∃𝑓𝑝(𝑧 = ⟨𝑓, 𝑝⟩ ∧ (𝑓(𝑉𝑊𝐸)𝑝𝜓)) → ∃𝑓𝑝(𝑧 = ⟨𝑓, 𝑝⟩ ∧ 𝜃)))
98ss2abdv 3265 . . 3 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → {𝑧 ∣ ∃𝑓𝑝(𝑧 = ⟨𝑓, 𝑝⟩ ∧ (𝑓(𝑉𝑊𝐸)𝑝𝜓))} ⊆ {𝑧 ∣ ∃𝑓𝑝(𝑧 = ⟨𝑓, 𝑝⟩ ∧ 𝜃)})
104, 9eqsstrid 3238 . 2 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑉𝑊𝐸)𝑝𝜓)} ⊆ {𝑧 ∣ ∃𝑓𝑝(𝑧 = ⟨𝑓, 𝑝⟩ ∧ 𝜃)})
113, 10ssexd 4183 1 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑉𝑊𝐸)𝑝𝜓)} ∈ V)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1372  wex 1514  wcel 2175  {cab 2190  Vcvv 2771  cop 3635   class class class wbr 4043  {copab 4103  (class class class)co 5943
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 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186  ax-sep 4161
This theorem depends on definitions:  df-bi 117  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-v 2773  df-in 3171  df-ss 3178  df-opab 4105
This theorem is referenced by: (None)
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