Theorem fnun

Description: The union of two functions with disjoint domains. (Contributed by NM, 22-Sep-2004)

	Ref	Expression
Assertion	fnun	⊢ ( ( ( 𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ) ∧ ( 𝐴 ∩ 𝐵 ) = ∅ ) → ( 𝐹 ∪ 𝐺 ) Fn ( 𝐴 ∪ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		df-fn	⊢ ( 𝐹 Fn 𝐴 ↔ ( Fun 𝐹 ∧ dom 𝐹 = 𝐴 ) )
2		df-fn	⊢ ( 𝐺 Fn 𝐵 ↔ ( Fun 𝐺 ∧ dom 𝐺 = 𝐵 ) )
3		ineq12	⊢ ( ( dom 𝐹 = 𝐴 ∧ dom 𝐺 = 𝐵 ) → ( dom 𝐹 ∩ dom 𝐺 ) = ( 𝐴 ∩ 𝐵 ) )
4	3	eqeq1d	⊢ ( ( dom 𝐹 = 𝐴 ∧ dom 𝐺 = 𝐵 ) → ( ( dom 𝐹 ∩ dom 𝐺 ) = ∅ ↔ ( 𝐴 ∩ 𝐵 ) = ∅ ) )
5	4	anbi2d	⊢ ( ( dom 𝐹 = 𝐴 ∧ dom 𝐺 = 𝐵 ) → ( ( ( Fun 𝐹 ∧ Fun 𝐺 ) ∧ ( dom 𝐹 ∩ dom 𝐺 ) = ∅ ) ↔ ( ( Fun 𝐹 ∧ Fun 𝐺 ) ∧ ( 𝐴 ∩ 𝐵 ) = ∅ ) ) )
6		funun	⊢ ( ( ( Fun 𝐹 ∧ Fun 𝐺 ) ∧ ( dom 𝐹 ∩ dom 𝐺 ) = ∅ ) → Fun ( 𝐹 ∪ 𝐺 ) )
7	5 6	biimtrrdi	⊢ ( ( dom 𝐹 = 𝐴 ∧ dom 𝐺 = 𝐵 ) → ( ( ( Fun 𝐹 ∧ Fun 𝐺 ) ∧ ( 𝐴 ∩ 𝐵 ) = ∅ ) → Fun ( 𝐹 ∪ 𝐺 ) ) )
8		dmun	⊢ dom ( 𝐹 ∪ 𝐺 ) = ( dom 𝐹 ∪ dom 𝐺 )
9		uneq12	⊢ ( ( dom 𝐹 = 𝐴 ∧ dom 𝐺 = 𝐵 ) → ( dom 𝐹 ∪ dom 𝐺 ) = ( 𝐴 ∪ 𝐵 ) )
10	8 9	eqtrid	⊢ ( ( dom 𝐹 = 𝐴 ∧ dom 𝐺 = 𝐵 ) → dom ( 𝐹 ∪ 𝐺 ) = ( 𝐴 ∪ 𝐵 ) )
11	7 10	jctird	⊢ ( ( dom 𝐹 = 𝐴 ∧ dom 𝐺 = 𝐵 ) → ( ( ( Fun 𝐹 ∧ Fun 𝐺 ) ∧ ( 𝐴 ∩ 𝐵 ) = ∅ ) → ( Fun ( 𝐹 ∪ 𝐺 ) ∧ dom ( 𝐹 ∪ 𝐺 ) = ( 𝐴 ∪ 𝐵 ) ) ) )
12		df-fn	⊢ ( ( 𝐹 ∪ 𝐺 ) Fn ( 𝐴 ∪ 𝐵 ) ↔ ( Fun ( 𝐹 ∪ 𝐺 ) ∧ dom ( 𝐹 ∪ 𝐺 ) = ( 𝐴 ∪ 𝐵 ) ) )
13	11 12	imbitrrdi	⊢ ( ( dom 𝐹 = 𝐴 ∧ dom 𝐺 = 𝐵 ) → ( ( ( Fun 𝐹 ∧ Fun 𝐺 ) ∧ ( 𝐴 ∩ 𝐵 ) = ∅ ) → ( 𝐹 ∪ 𝐺 ) Fn ( 𝐴 ∪ 𝐵 ) ) )
14	13	expd	⊢ ( ( dom 𝐹 = 𝐴 ∧ dom 𝐺 = 𝐵 ) → ( ( Fun 𝐹 ∧ Fun 𝐺 ) → ( ( 𝐴 ∩ 𝐵 ) = ∅ → ( 𝐹 ∪ 𝐺 ) Fn ( 𝐴 ∪ 𝐵 ) ) ) )
15	14	impcom	⊢ ( ( ( Fun 𝐹 ∧ Fun 𝐺 ) ∧ ( dom 𝐹 = 𝐴 ∧ dom 𝐺 = 𝐵 ) ) → ( ( 𝐴 ∩ 𝐵 ) = ∅ → ( 𝐹 ∪ 𝐺 ) Fn ( 𝐴 ∪ 𝐵 ) ) )
16	15	an4s	⊢ ( ( ( Fun 𝐹 ∧ dom 𝐹 = 𝐴 ) ∧ ( Fun 𝐺 ∧ dom 𝐺 = 𝐵 ) ) → ( ( 𝐴 ∩ 𝐵 ) = ∅ → ( 𝐹 ∪ 𝐺 ) Fn ( 𝐴 ∪ 𝐵 ) ) )
17	1 2 16	syl2anb	⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ) → ( ( 𝐴 ∩ 𝐵 ) = ∅ → ( 𝐹 ∪ 𝐺 ) Fn ( 𝐴 ∪ 𝐵 ) ) )
18	17	imp	⊢ ( ( ( 𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ) ∧ ( 𝐴 ∩ 𝐵 ) = ∅ ) → ( 𝐹 ∪ 𝐺 ) Fn ( 𝐴 ∪ 𝐵 ) )