Theorem elmapresaunres2

Description: fresaunres2 transposed to mappings. (Contributed by Stefan O'Rear, 9-Oct-2014)

	Ref	Expression
Assertion	elmapresaunres2	⊢ ( ( 𝐹 ∈ ( 𝐶 ↑m 𝐴 ) ∧ 𝐺 ∈ ( 𝐶 ↑m 𝐵 ) ∧ ( 𝐹 ↾ ( 𝐴 ∩ 𝐵 ) ) = ( 𝐺 ↾ ( 𝐴 ∩ 𝐵 ) ) ) → ( ( 𝐹 ∪ 𝐺 ) ↾ 𝐵 ) = 𝐺 )

Proof

Step	Hyp	Ref	Expression
1		elmapi	⊢ ( 𝐹 ∈ ( 𝐶 ↑m 𝐴 ) → 𝐹 : 𝐴 ⟶ 𝐶 )
2		elmapi	⊢ ( 𝐺 ∈ ( 𝐶 ↑m 𝐵 ) → 𝐺 : 𝐵 ⟶ 𝐶 )
3		id	⊢ ( ( 𝐹 ↾ ( 𝐴 ∩ 𝐵 ) ) = ( 𝐺 ↾ ( 𝐴 ∩ 𝐵 ) ) → ( 𝐹 ↾ ( 𝐴 ∩ 𝐵 ) ) = ( 𝐺 ↾ ( 𝐴 ∩ 𝐵 ) ) )
4		fresaunres2	⊢ ( ( 𝐹 : 𝐴 ⟶ 𝐶 ∧ 𝐺 : 𝐵 ⟶ 𝐶 ∧ ( 𝐹 ↾ ( 𝐴 ∩ 𝐵 ) ) = ( 𝐺 ↾ ( 𝐴 ∩ 𝐵 ) ) ) → ( ( 𝐹 ∪ 𝐺 ) ↾ 𝐵 ) = 𝐺 )
5	1 2 3 4	syl3an	⊢ ( ( 𝐹 ∈ ( 𝐶 ↑m 𝐴 ) ∧ 𝐺 ∈ ( 𝐶 ↑m 𝐵 ) ∧ ( 𝐹 ↾ ( 𝐴 ∩ 𝐵 ) ) = ( 𝐺 ↾ ( 𝐴 ∩ 𝐵 ) ) ) → ( ( 𝐹 ∪ 𝐺 ) ↾ 𝐵 ) = 𝐺 )