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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 ⊢ ( 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 ) ) ) )