Description: A singleton index picks out an instance of an indexed union's argument. (Contributed by Mario Carneiro, 25-Jun-2016) (Revised by Thierry Arnoux, 2-May-2020) Avoid ax-13 . (Revised by Gino Giotto, 19-May-2023)
| Ref | Expression | ||
|---|---|---|---|
| Hypotheses | iunxsngf.1 | ⊢ Ⅎ 𝑥 𝐶 | |
| iunxsngf.2 | ⊢ ( 𝑥 = 𝐴 → 𝐵 = 𝐶 ) | ||
| Assertion | iunxsngf | ⊢ ( 𝐴 ∈ 𝑉 → ∪ 𝑥 ∈ { 𝐴 } 𝐵 = 𝐶 ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | iunxsngf.1 | ⊢ Ⅎ 𝑥 𝐶 | |
| 2 | iunxsngf.2 | ⊢ ( 𝑥 = 𝐴 → 𝐵 = 𝐶 ) | |
| 3 | eliun | ⊢ ( 𝑦 ∈ ∪ 𝑥 ∈ { 𝐴 } 𝐵 ↔ ∃ 𝑥 ∈ { 𝐴 } 𝑦 ∈ 𝐵 ) | |
| 4 | 1 | nfcri | ⊢ Ⅎ 𝑥 𝑦 ∈ 𝐶 |
| 5 | 2 | eleq2d | ⊢ ( 𝑥 = 𝐴 → ( 𝑦 ∈ 𝐵 ↔ 𝑦 ∈ 𝐶 ) ) |
| 6 | 4 5 | rexsngf | ⊢ ( 𝐴 ∈ 𝑉 → ( ∃ 𝑥 ∈ { 𝐴 } 𝑦 ∈ 𝐵 ↔ 𝑦 ∈ 𝐶 ) ) |
| 7 | 3 6 | syl5bb | ⊢ ( 𝐴 ∈ 𝑉 → ( 𝑦 ∈ ∪ 𝑥 ∈ { 𝐴 } 𝐵 ↔ 𝑦 ∈ 𝐶 ) ) |
| 8 | 7 | eqrdv | ⊢ ( 𝐴 ∈ 𝑉 → ∪ 𝑥 ∈ { 𝐴 } 𝐵 = 𝐶 ) |