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Metamath Proof Explorer

Theorem oprabex3

Description: Existence of an operation class abstraction (special case). (Contributed by NM, 19-Oct-2004)

Ref Expression
Hypotheses oprabex3.1 ⊢ 𝐻 ∈ V
oprabex3.2 ⊢ 𝐹 = { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ ( ( 𝑥 ∈ ( 𝐻 × 𝐻 ) ∧ 𝑦 ∈ ( 𝐻 × 𝐻 ) ) ∧ ∃ 𝑤 ∃ 𝑣 ∃ 𝑢 ∃ 𝑓 ( ( 𝑥 = ⟨ 𝑤 , 𝑣 ⟩ ∧ 𝑦 = ⟨ 𝑢 , 𝑓 ⟩ ) ∧ 𝑧 = 𝑅 ) ) }
Assertion oprabex3 𝐹 ∈ V

Proof

Step Hyp Ref Expression
1 oprabex3.1 ⊢ 𝐻 ∈ V
2 oprabex3.2 ⊢ 𝐹 = { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ ( ( 𝑥 ∈ ( 𝐻 × 𝐻 ) ∧ 𝑦 ∈ ( 𝐻 × 𝐻 ) ) ∧ ∃ 𝑤 ∃ 𝑣 ∃ 𝑢 ∃ 𝑓 ( ( 𝑥 = ⟨ 𝑤 , 𝑣 ⟩ ∧ 𝑦 = ⟨ 𝑢 , 𝑓 ⟩ ) ∧ 𝑧 = 𝑅 ) ) }
3 1 1 xpex ⊢ ( 𝐻 × 𝐻 ) ∈ V
4 moeq ⊢ ∃* 𝑧 𝑧 = 𝑅
5 4 mosubop ⊢ ∃* 𝑧 ∃ 𝑢 ∃ 𝑓 ( 𝑦 = ⟨ 𝑢 , 𝑓 ⟩ ∧ 𝑧 = 𝑅 )
6 5 mosubop ⊢ ∃* 𝑧 ∃ 𝑤 ∃ 𝑣 ( 𝑥 = ⟨ 𝑤 , 𝑣 ⟩ ∧ ∃ 𝑢 ∃ 𝑓 ( 𝑦 = ⟨ 𝑢 , 𝑓 ⟩ ∧ 𝑧 = 𝑅 ) )
7 anass ⊢ ( ( ( 𝑥 = ⟨ 𝑤 , 𝑣 ⟩ ∧ 𝑦 = ⟨ 𝑢 , 𝑓 ⟩ ) ∧ 𝑧 = 𝑅 ) ↔ ( 𝑥 = ⟨ 𝑤 , 𝑣 ⟩ ∧ ( 𝑦 = ⟨ 𝑢 , 𝑓 ⟩ ∧ 𝑧 = 𝑅 ) ) )
8 7 2exbii ⊢ ( ∃ 𝑢 ∃ 𝑓 ( ( 𝑥 = ⟨ 𝑤 , 𝑣 ⟩ ∧ 𝑦 = ⟨ 𝑢 , 𝑓 ⟩ ) ∧ 𝑧 = 𝑅 ) ↔ ∃ 𝑢 ∃ 𝑓 ( 𝑥 = ⟨ 𝑤 , 𝑣 ⟩ ∧ ( 𝑦 = ⟨ 𝑢 , 𝑓 ⟩ ∧ 𝑧 = 𝑅 ) ) )
9 19.42vv ⊢ ( ∃ 𝑢 ∃ 𝑓 ( 𝑥 = ⟨ 𝑤 , 𝑣 ⟩ ∧ ( 𝑦 = ⟨ 𝑢 , 𝑓 ⟩ ∧ 𝑧 = 𝑅 ) ) ↔ ( 𝑥 = ⟨ 𝑤 , 𝑣 ⟩ ∧ ∃ 𝑢 ∃ 𝑓 ( 𝑦 = ⟨ 𝑢 , 𝑓 ⟩ ∧ 𝑧 = 𝑅 ) ) )
10 8 9 bitri ⊢ ( ∃ 𝑢 ∃ 𝑓 ( ( 𝑥 = ⟨ 𝑤 , 𝑣 ⟩ ∧ 𝑦 = ⟨ 𝑢 , 𝑓 ⟩ ) ∧ 𝑧 = 𝑅 ) ↔ ( 𝑥 = ⟨ 𝑤 , 𝑣 ⟩ ∧ ∃ 𝑢 ∃ 𝑓 ( 𝑦 = ⟨ 𝑢 , 𝑓 ⟩ ∧ 𝑧 = 𝑅 ) ) )
11 10 2exbii ⊢ ( ∃ 𝑤 ∃ 𝑣 ∃ 𝑢 ∃ 𝑓 ( ( 𝑥 = ⟨ 𝑤 , 𝑣 ⟩ ∧ 𝑦 = ⟨ 𝑢 , 𝑓 ⟩ ) ∧ 𝑧 = 𝑅 ) ↔ ∃ 𝑤 ∃ 𝑣 ( 𝑥 = ⟨ 𝑤 , 𝑣 ⟩ ∧ ∃ 𝑢 ∃ 𝑓 ( 𝑦 = ⟨ 𝑢 , 𝑓 ⟩ ∧ 𝑧 = 𝑅 ) ) )
12 11 mobii ⊢ ( ∃* 𝑧 ∃ 𝑤 ∃ 𝑣 ∃ 𝑢 ∃ 𝑓 ( ( 𝑥 = ⟨ 𝑤 , 𝑣 ⟩ ∧ 𝑦 = ⟨ 𝑢 , 𝑓 ⟩ ) ∧ 𝑧 = 𝑅 ) ↔ ∃* 𝑧 ∃ 𝑤 ∃ 𝑣 ( 𝑥 = ⟨ 𝑤 , 𝑣 ⟩ ∧ ∃ 𝑢 ∃ 𝑓 ( 𝑦 = ⟨ 𝑢 , 𝑓 ⟩ ∧ 𝑧 = 𝑅 ) ) )
13 6 12 mpbir ⊢ ∃* 𝑧 ∃ 𝑤 ∃ 𝑣 ∃ 𝑢 ∃ 𝑓 ( ( 𝑥 = ⟨ 𝑤 , 𝑣 ⟩ ∧ 𝑦 = ⟨ 𝑢 , 𝑓 ⟩ ) ∧ 𝑧 = 𝑅 )
14 13 a1i ⊢ ( ( 𝑥 ∈ ( 𝐻 × 𝐻 ) ∧ 𝑦 ∈ ( 𝐻 × 𝐻 ) ) → ∃* 𝑧 ∃ 𝑤 ∃ 𝑣 ∃ 𝑢 ∃ 𝑓 ( ( 𝑥 = ⟨ 𝑤 , 𝑣 ⟩ ∧ 𝑦 = ⟨ 𝑢 , 𝑓 ⟩ ) ∧ 𝑧 = 𝑅 ) )
15 3 3 14 2 oprabex ⊢ 𝐹 ∈ V