Theorem bnj1383

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011) (New usage is discouraged.)

	Ref	Expression
Hypotheses	bnj1383.1	⊢ ( 𝜑 ↔ ∀ 𝑓 ∈ 𝐴 Fun 𝑓 )
	bnj1383.2	⊢ 𝐷 = ( dom 𝑓 ∩ dom 𝑔 )
	bnj1383.3	⊢ ( 𝜓 ↔ ( 𝜑 ∧ ∀ 𝑓 ∈ 𝐴 ∀ 𝑔 ∈ 𝐴 ( 𝑓 ↾ 𝐷 ) = ( 𝑔 ↾ 𝐷 ) ) )
Assertion	bnj1383	⊢ ( 𝜓 → Fun ∪ 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		bnj1383.1	⊢ ( 𝜑 ↔ ∀ 𝑓 ∈ 𝐴 Fun 𝑓 )
2		bnj1383.2	⊢ 𝐷 = ( dom 𝑓 ∩ dom 𝑔 )
3		bnj1383.3	⊢ ( 𝜓 ↔ ( 𝜑 ∧ ∀ 𝑓 ∈ 𝐴 ∀ 𝑔 ∈ 𝐴 ( 𝑓 ↾ 𝐷 ) = ( 𝑔 ↾ 𝐷 ) ) )
4		biid	⊢ ( ( 𝜓 ∧ ⟨ 𝑥 , 𝑦 ⟩ ∈ ∪ 𝐴 ∧ ⟨ 𝑥 , 𝑧 ⟩ ∈ ∪ 𝐴 ) ↔ ( 𝜓 ∧ ⟨ 𝑥 , 𝑦 ⟩ ∈ ∪ 𝐴 ∧ ⟨ 𝑥 , 𝑧 ⟩ ∈ ∪ 𝐴 ) )
5		biid	⊢ ( ( ( 𝜓 ∧ ⟨ 𝑥 , 𝑦 ⟩ ∈ ∪ 𝐴 ∧ ⟨ 𝑥 , 𝑧 ⟩ ∈ ∪ 𝐴 ) ∧ 𝑓 ∈ 𝐴 ∧ ⟨ 𝑥 , 𝑦 ⟩ ∈ 𝑓 ) ↔ ( ( 𝜓 ∧ ⟨ 𝑥 , 𝑦 ⟩ ∈ ∪ 𝐴 ∧ ⟨ 𝑥 , 𝑧 ⟩ ∈ ∪ 𝐴 ) ∧ 𝑓 ∈ 𝐴 ∧ ⟨ 𝑥 , 𝑦 ⟩ ∈ 𝑓 ) )
6		biid	⊢ ( ( ( ( 𝜓 ∧ ⟨ 𝑥 , 𝑦 ⟩ ∈ ∪ 𝐴 ∧ ⟨ 𝑥 , 𝑧 ⟩ ∈ ∪ 𝐴 ) ∧ 𝑓 ∈ 𝐴 ∧ ⟨ 𝑥 , 𝑦 ⟩ ∈ 𝑓 ) ∧ 𝑔 ∈ 𝐴 ∧ ⟨ 𝑥 , 𝑧 ⟩ ∈ 𝑔 ) ↔ ( ( ( 𝜓 ∧ ⟨ 𝑥 , 𝑦 ⟩ ∈ ∪ 𝐴 ∧ ⟨ 𝑥 , 𝑧 ⟩ ∈ ∪ 𝐴 ) ∧ 𝑓 ∈ 𝐴 ∧ ⟨ 𝑥 , 𝑦 ⟩ ∈ 𝑓 ) ∧ 𝑔 ∈ 𝐴 ∧ ⟨ 𝑥 , 𝑧 ⟩ ∈ 𝑔 ) )
7	1 2 3 4 5 6	bnj1379	⊢ ( 𝜓 → Fun ∪ 𝐴 )