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Theorem opabbrex 5781
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 3958 . . 3 {⟨𝑓, 𝑝⟩ ∣ 𝜃} = {𝑧 ∣ ∃𝑓𝑝(𝑧 = ⟨𝑓, 𝑝⟩ ∧ 𝜃)}
2 opabbrex.2 . . 3 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → {⟨𝑓, 𝑝⟩ ∣ 𝜃} ∈ V)
31, 2eqeltrrid 2203 . 2 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → {𝑧 ∣ ∃𝑓𝑝(𝑧 = ⟨𝑓, 𝑝⟩ ∧ 𝜃)} ∈ V)
4 df-opab 3958 . . 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 275 . . . . . 6 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → ((𝑓(𝑉𝑊𝐸)𝑝𝜓) → 𝜃))
76anim2d 333 . . . . 5 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → ((𝑧 = ⟨𝑓, 𝑝⟩ ∧ (𝑓(𝑉𝑊𝐸)𝑝𝜓)) → (𝑧 = ⟨𝑓, 𝑝⟩ ∧ 𝜃)))
872eximdv 1836 . . . 4 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (∃𝑓𝑝(𝑧 = ⟨𝑓, 𝑝⟩ ∧ (𝑓(𝑉𝑊𝐸)𝑝𝜓)) → ∃𝑓𝑝(𝑧 = ⟨𝑓, 𝑝⟩ ∧ 𝜃)))
98ss2abdv 3138 . . 3 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → {𝑧 ∣ ∃𝑓𝑝(𝑧 = ⟨𝑓, 𝑝⟩ ∧ (𝑓(𝑉𝑊𝐸)𝑝𝜓))} ⊆ {𝑧 ∣ ∃𝑓𝑝(𝑧 = ⟨𝑓, 𝑝⟩ ∧ 𝜃)})
104, 9eqsstrid 3111 . 2 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑉𝑊𝐸)𝑝𝜓)} ⊆ {𝑧 ∣ ∃𝑓𝑝(𝑧 = ⟨𝑓, 𝑝⟩ ∧ 𝜃)})
113, 10ssexd 4036 1 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑉𝑊𝐸)𝑝𝜓)} ∈ V)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1314  wex 1451  wcel 1463  {cab 2101  Vcvv 2658  cop 3498   class class class wbr 3897  {copab 3956  (class class class)co 5740
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-v 2660  df-in 3045  df-ss 3052  df-opab 3958
This theorem is referenced by: (None)
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