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Metamath Proof Explorer

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 )