Description: Appending a set to an indexed union. (Contributed by Thierry Arnoux, 20-Nov-2023)
Ref | Expression | ||
---|---|---|---|
Hypothesis | iunxunsn.1 | ⊢ ( 𝑥 = 𝑋 → 𝐵 = 𝐶 ) | |
Assertion | iunxunsn | ⊢ ( 𝑋 ∈ 𝑉 → ∪ 𝑥 ∈ ( 𝐴 ∪ { 𝑋 } ) 𝐵 = ( ∪ 𝑥 ∈ 𝐴 𝐵 ∪ 𝐶 ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iunxunsn.1 | ⊢ ( 𝑥 = 𝑋 → 𝐵 = 𝐶 ) | |
2 | iunxun | ⊢ ∪ 𝑥 ∈ ( 𝐴 ∪ { 𝑋 } ) 𝐵 = ( ∪ 𝑥 ∈ 𝐴 𝐵 ∪ ∪ 𝑥 ∈ { 𝑋 } 𝐵 ) | |
3 | 1 | iunxsng | ⊢ ( 𝑋 ∈ 𝑉 → ∪ 𝑥 ∈ { 𝑋 } 𝐵 = 𝐶 ) |
4 | 3 | uneq2d | ⊢ ( 𝑋 ∈ 𝑉 → ( ∪ 𝑥 ∈ 𝐴 𝐵 ∪ ∪ 𝑥 ∈ { 𝑋 } 𝐵 ) = ( ∪ 𝑥 ∈ 𝐴 𝐵 ∪ 𝐶 ) ) |
5 | 2 4 | syl5eq | ⊢ ( 𝑋 ∈ 𝑉 → ∪ 𝑥 ∈ ( 𝐴 ∪ { 𝑋 } ) 𝐵 = ( ∪ 𝑥 ∈ 𝐴 𝐵 ∪ 𝐶 ) ) |