Theorem opabbrex 5549

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 3819	. . 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	syl5eqelr 2125	. 2 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → {𝑧 ∣ ∃𝑓∃𝑝(𝑧 = ⟨𝑓, 𝑝⟩ ∧ 𝜃)} ∈ V)
4		df-opab 3819	. . 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 264	. . . . . 6 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → ((𝑓(𝑉𝑊𝐸)𝑝 ∧ 𝜓) → 𝜃))
7	6	anim2d 320	. . . . 5 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → ((𝑧 = ⟨𝑓, 𝑝⟩ ∧ (𝑓(𝑉𝑊𝐸)𝑝 ∧ 𝜓)) → (𝑧 = ⟨𝑓, 𝑝⟩ ∧ 𝜃)))
8	7	2eximdv 1762	. . . 4 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (∃𝑓∃𝑝(𝑧 = ⟨𝑓, 𝑝⟩ ∧ (𝑓(𝑉𝑊𝐸)𝑝 ∧ 𝜓)) → ∃𝑓∃𝑝(𝑧 = ⟨𝑓, 𝑝⟩ ∧ 𝜃)))
9	8	ss2abdv 3013	. . 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	syl5eqss 2989	. 2 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑉𝑊𝐸)𝑝 ∧ 𝜓)} ⊆ {𝑧 ∣ ∃𝑓∃𝑝(𝑧 = ⟨𝑓, 𝑝⟩ ∧ 𝜃)})
11	3, 10	ssexd 3897	1 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑉𝑊𝐸)𝑝 ∧ 𝜓)} ∈ V)

Syntax hints:   → wi 4   ∧ wa 97   = wceq 1243  ∃wex 1381   ∈ wcel 1393  {cab 2026  Vcvv 2557  ⟨cop 3378   class class class wbr 3764  {copab 3817  (class class class)co 5512

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-in 2924  df-ss 2931  df-opab 3819

This theorem is referenced by:  sprmpt2  5857