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 ) ) ) ) |
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 | syl6bi | ⊢ ( ( 𝜑 ∧ ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉 ) ) → ( ( ( 𝐹 ‘ 𝐴 ) = ( 𝐹 ‘ 𝐶 ) ∧ ( 𝐹 ‘ 𝐵 ) = ( 𝐹 ‘ 𝐷 ) ) → ( 𝐹 ‘ ( 𝐴 + 𝐵 ) ) = ( 𝐹 ‘ ( 𝐶 + 𝐷 ) ) ) ) |