Metamath Proof Explorer


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 ( 𝜑𝐹 : 𝑉1-1-onto𝑋 )
Assertion f1ocpbl ( ( 𝜑 ∧ ( 𝐴𝑉𝐵𝑉 ) ∧ ( 𝐶𝑉𝐷𝑉 ) ) → ( ( ( 𝐹𝐴 ) = ( 𝐹𝐶 ) ∧ ( 𝐹𝐵 ) = ( 𝐹𝐷 ) ) → ( 𝐹 ‘ ( 𝐴 + 𝐵 ) ) = ( 𝐹 ‘ ( 𝐶 + 𝐷 ) ) ) )

Proof

Step Hyp Ref Expression
1 f1ocpbl.f ( 𝜑𝐹 : 𝑉1-1-onto𝑋 )
2 1 f1ocpbllem ( ( 𝜑 ∧ ( 𝐴𝑉𝐵𝑉 ) ∧ ( 𝐶𝑉𝐷𝑉 ) ) → ( ( ( 𝐹𝐴 ) = ( 𝐹𝐶 ) ∧ ( 𝐹𝐵 ) = ( 𝐹𝐷 ) ) ↔ ( 𝐴 = 𝐶𝐵 = 𝐷 ) ) )
3 oveq12 ( ( 𝐴 = 𝐶𝐵 = 𝐷 ) → ( 𝐴 + 𝐵 ) = ( 𝐶 + 𝐷 ) )
4 3 fveq2d ( ( 𝐴 = 𝐶𝐵 = 𝐷 ) → ( 𝐹 ‘ ( 𝐴 + 𝐵 ) ) = ( 𝐹 ‘ ( 𝐶 + 𝐷 ) ) )
5 2 4 syl6bi ( ( 𝜑 ∧ ( 𝐴𝑉𝐵𝑉 ) ∧ ( 𝐶𝑉𝐷𝑉 ) ) → ( ( ( 𝐹𝐴 ) = ( 𝐹𝐶 ) ∧ ( 𝐹𝐵 ) = ( 𝐹𝐷 ) ) → ( 𝐹 ‘ ( 𝐴 + 𝐵 ) ) = ( 𝐹 ‘ ( 𝐶 + 𝐷 ) ) ) )