Theorem fnunsn

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

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

Proof

Step	Hyp	Ref	Expression
1		fnunop.x	⊢ ( 𝜑 → 𝑋 ∈ V )
2		fnunop.y	⊢ ( 𝜑 → 𝑌 ∈ V )
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 ∈ V ) → { ⟨ 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		fnun	⊢ ( ( ( 𝐹 Fn 𝐷 ∧ { ⟨ 𝑋 , 𝑌 ⟩ } Fn { 𝑋 } ) ∧ ( 𝐷 ∩ { 𝑋 } ) = ∅ ) → ( 𝐹 ∪ { ⟨ 𝑋 , 𝑌 ⟩ } ) Fn ( 𝐷 ∪ { 𝑋 } ) )
12	3 8 10 11	syl21anc	⊢ ( 𝜑 → ( 𝐹 ∪ { ⟨ 𝑋 , 𝑌 ⟩ } ) Fn ( 𝐷 ∪ { 𝑋 } ) )
13	4	fneq1i	⊢ ( 𝐺 Fn 𝐸 ↔ ( 𝐹 ∪ { ⟨ 𝑋 , 𝑌 ⟩ } ) Fn 𝐸 )
14	5	fneq2i	⊢ ( ( 𝐹 ∪ { ⟨ 𝑋 , 𝑌 ⟩ } ) Fn 𝐸 ↔ ( 𝐹 ∪ { ⟨ 𝑋 , 𝑌 ⟩ } ) Fn ( 𝐷 ∪ { 𝑋 } ) )
15	13 14	bitri	⊢ ( 𝐺 Fn 𝐸 ↔ ( 𝐹 ∪ { ⟨ 𝑋 , 𝑌 ⟩ } ) Fn ( 𝐷 ∪ { 𝑋 } ) )
16	12 15	sylibr	⊢ ( 𝜑 → 𝐺 Fn 𝐸 )