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Metamath Proof Explorer

Theorem fnunop

Description: Extension of a function with a new ordered pair. (Contributed by NM, 28-Sep-2013) (Revised by Mario Carneiro, 30-Apr-2015) (Revised by AV, 16-Aug-2024)

Ref Expression
Hypotheses fnunop.x ⊢ ( 𝜑 → 𝑋 ∈ 𝑉 )
fnunop.y ⊢ ( 𝜑 → 𝑌 ∈ 𝑊 )
fnunop.f ⊢ ( 𝜑 → 𝐹 Fn 𝐷 )
fnunop.g ⊢ 𝐺 = ( 𝐹 ∪ { ⟨ 𝑋 , 𝑌 ⟩ } )
fnunop.e ⊢ 𝐸 = ( 𝐷 ∪ { 𝑋 } )
fnunop.d ⊢ ( 𝜑 → ¬ 𝑋 ∈ 𝐷 )
Assertion fnunop ( 𝜑 → 𝐺 Fn 𝐸 )

Proof

Step Hyp Ref Expression
1 fnunop.x ⊢ ( 𝜑 → 𝑋 ∈ 𝑉 )
2 fnunop.y ⊢ ( 𝜑 → 𝑌 ∈ 𝑊 )
3 fnunop.f ⊢ ( 𝜑 → 𝐹 Fn 𝐷 )
4 fnunop.g ⊢ 𝐺 = ( 𝐹 ∪ { ⟨ 𝑋 , 𝑌 ⟩ } )
5 fnunop.e ⊢ 𝐸 = ( 𝐷 ∪ { 𝑋 } )
6 fnunop.d ⊢ ( 𝜑 → ¬ 𝑋 ∈ 𝐷 )
7 fnsng ⊢ ( ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ) → { ⟨ 𝑋 , 𝑌 ⟩ } Fn { 𝑋 } )
8 1 2 7 syl2anc ⊢ ( 𝜑 → { ⟨ 𝑋 , 𝑌 ⟩ } Fn { 𝑋 } )
9 disjsn ⊢ ( ( 𝐷 ∩ { 𝑋 } ) = ∅ ↔ ¬ 𝑋 ∈ 𝐷 )
10 6 9 sylibr ⊢ ( 𝜑 → ( 𝐷 ∩ { 𝑋 } ) = ∅ )
11 3 8 10 fnund ⊢ ( 𝜑 → ( 𝐹 ∪ { ⟨ 𝑋 , 𝑌 ⟩ } ) Fn ( 𝐷 ∪ { 𝑋 } ) )
12 4 fneq1i ⊢ ( 𝐺 Fn 𝐸 ↔ ( 𝐹 ∪ { ⟨ 𝑋 , 𝑌 ⟩ } ) Fn 𝐸 )
13 5 fneq2i ⊢ ( ( 𝐹 ∪ { ⟨ 𝑋 , 𝑌 ⟩ } ) Fn 𝐸 ↔ ( 𝐹 ∪ { ⟨ 𝑋 , 𝑌 ⟩ } ) Fn ( 𝐷 ∪ { 𝑋 } ) )
14 12 13 bitri ⊢ ( 𝐺 Fn 𝐸 ↔ ( 𝐹 ∪ { ⟨ 𝑋 , 𝑌 ⟩ } ) Fn ( 𝐷 ∪ { 𝑋 } ) )
15 11 14 sylibr ⊢ ( 𝜑 → 𝐺 Fn 𝐸 )