Theorem opabbrex 5932

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)𝑝 → 𝜃))
opabbrex.2	⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → {⟨𝑓, 𝑝⟩ ∣ 𝜃} ∈ V)

Assertion

Ref	Expression
opabbrex	⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑉𝑊𝐸)𝑝 ∧ 𝜓)} ∈ 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.

Step	Hyp	Ref	Expression
1		df-opab 4077	. . 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 2275	. 2 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → {𝑧 ∣ ∃𝑓∃𝑝(𝑧 = ⟨𝑓, 𝑝⟩ ∧ 𝜃)} ∈ V)
4		df-opab 4077	. . 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 1892	. . . 4 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (∃𝑓∃𝑝(𝑧 = ⟨𝑓, 𝑝⟩ ∧ (𝑓(𝑉𝑊𝐸)𝑝 ∧ 𝜓)) → ∃𝑓∃𝑝(𝑧 = ⟨𝑓, 𝑝⟩ ∧ 𝜃)))
9	8	ss2abdv 3240	. . 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 3213	. 2 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑉𝑊𝐸)𝑝 ∧ 𝜓)} ⊆ {𝑧 ∣ ∃𝑓∃𝑝(𝑧 = ⟨𝑓, 𝑝⟩ ∧ 𝜃)})
11	3, 10	ssexd 4155	1 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑉𝑊𝐸)𝑝 ∧ 𝜓)} ∈ V)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1363  ∃wex 1502   ∈ wcel 2158  {cab 2173  Vcvv 2749  ⟨cop 3607   class class class wbr 4015  {copab 4075  (class class class)co 5888

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169  ax-sep 4133

This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-v 2751  df-in 3147  df-ss 3154  df-opab 4077

This theorem is referenced by: (None)