Theorem f1ocpbl

Description: An injection is compatible with any operations on the base set. (Contributed by Mario Carneiro, 24-Feb-2015)

	Ref	Expression
Hypothesis	f1ocpbl.f	⊢ ( 𝜑 → 𝐹 : 𝑉 –1-1-onto→ 𝑋 )
Assertion	f1ocpbl	⊢ ( ( 𝜑 ∧ ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉 ) ) → ( ( ( 𝐹 ‘ 𝐴 ) = ( 𝐹 ‘ 𝐶 ) ∧ ( 𝐹 ‘ 𝐵 ) = ( 𝐹 ‘ 𝐷 ) ) → ( 𝐹 ‘ ( 𝐴 + 𝐵 ) ) = ( 𝐹 ‘ ( 𝐶 + 𝐷 ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		f1ocpbl.f	⊢ ( 𝜑 → 𝐹 : 𝑉 –1-1-onto→ 𝑋 )
2	1	f1ocpbllem	⊢ ( ( 𝜑 ∧ ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉 ) ) → ( ( ( 𝐹 ‘ 𝐴 ) = ( 𝐹 ‘ 𝐶 ) ∧ ( 𝐹 ‘ 𝐵 ) = ( 𝐹 ‘ 𝐷 ) ) ↔ ( 𝐴 = 𝐶 ∧ 𝐵 = 𝐷 ) ) )
3		oveq12	⊢ ( ( 𝐴 = 𝐶 ∧ 𝐵 = 𝐷 ) → ( 𝐴 + 𝐵 ) = ( 𝐶 + 𝐷 ) )
4	3	fveq2d	⊢ ( ( 𝐴 = 𝐶 ∧ 𝐵 = 𝐷 ) → ( 𝐹 ‘ ( 𝐴 + 𝐵 ) ) = ( 𝐹 ‘ ( 𝐶 + 𝐷 ) ) )
5	2 4	biimtrdi	⊢ ( ( 𝜑 ∧ ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉 ) ) → ( ( ( 𝐹 ‘ 𝐴 ) = ( 𝐹 ‘ 𝐶 ) ∧ ( 𝐹 ‘ 𝐵 ) = ( 𝐹 ‘ 𝐷 ) ) → ( 𝐹 ‘ ( 𝐴 + 𝐵 ) ) = ( 𝐹 ‘ ( 𝐶 + 𝐷 ) ) ) )