Description: Value of a function expressed as a union of a function and a singleton on a couple (with disjoint domain) at a point not equal to the first component of that couple. (Contributed by BJ, 18-Mar-2023) (Proof modification is discouraged.)
| Ref | Expression | ||
|---|---|---|---|
| Hypotheses | bj-fununsn.un | ⊢ ( 𝜑 → 𝐹 = ( 𝐺 ∪ { ⟨ 𝐵 , 𝐶 ⟩ } ) ) | |
| bj-fununsn1.neq | ⊢ ( 𝜑 → ¬ 𝐴 = 𝐵 ) | ||
| Assertion | bj-fununsn1 | ⊢ ( 𝜑 → ( 𝐹 ‘ 𝐴 ) = ( 𝐺 ‘ 𝐴 ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | bj-fununsn.un | ⊢ ( 𝜑 → 𝐹 = ( 𝐺 ∪ { ⟨ 𝐵 , 𝐶 ⟩ } ) ) | |
| 2 | bj-fununsn1.neq | ⊢ ( 𝜑 → ¬ 𝐴 = 𝐵 ) | |
| 3 | dmsnopss | ⊢ dom { ⟨ 𝐵 , 𝐶 ⟩ } ⊆ { 𝐵 } | |
| 4 | 3 | a1i | ⊢ ( 𝜑 → dom { ⟨ 𝐵 , 𝐶 ⟩ } ⊆ { 𝐵 } ) |
| 5 | elsni | ⊢ ( 𝐴 ∈ { 𝐵 } → 𝐴 = 𝐵 ) | |
| 6 | 2 5 | nsyl | ⊢ ( 𝜑 → ¬ 𝐴 ∈ { 𝐵 } ) |
| 7 | 4 6 | ssneldd | ⊢ ( 𝜑 → ¬ 𝐴 ∈ dom { ⟨ 𝐵 , 𝐶 ⟩ } ) |
| 8 | 1 7 | bj-funun | ⊢ ( 𝜑 → ( 𝐹 ‘ 𝐴 ) = ( 𝐺 ‘ 𝐴 ) ) |