Description: A class B is equal to the union of the class of all singletons of elements of B . (Contributed by ML, 16-Jul-2020)
Ref | Expression | ||
---|---|---|---|
Hypotheses | mptsnun.f | ⊢ 𝐹 = ( 𝑥 ∈ 𝐴 ↦ { 𝑥 } ) | |
mptsnun.r | ⊢ 𝑅 = { 𝑢 ∣ ∃ 𝑥 ∈ 𝐴 𝑢 = { 𝑥 } } | ||
Assertion | mptsnun | ⊢ ( 𝐵 ⊆ 𝐴 → 𝐵 = ∪ ( 𝐹 “ 𝐵 ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mptsnun.f | ⊢ 𝐹 = ( 𝑥 ∈ 𝐴 ↦ { 𝑥 } ) | |
2 | mptsnun.r | ⊢ 𝑅 = { 𝑢 ∣ ∃ 𝑥 ∈ 𝐴 𝑢 = { 𝑥 } } | |
3 | sneq | ⊢ ( 𝑥 = 𝑦 → { 𝑥 } = { 𝑦 } ) | |
4 | 3 | cbvmptv | ⊢ ( 𝑥 ∈ 𝐴 ↦ { 𝑥 } ) = ( 𝑦 ∈ 𝐴 ↦ { 𝑦 } ) |
5 | 4 | eqcomi | ⊢ ( 𝑦 ∈ 𝐴 ↦ { 𝑦 } ) = ( 𝑥 ∈ 𝐴 ↦ { 𝑥 } ) |
6 | 5 2 | mptsnunlem | ⊢ ( 𝐵 ⊆ 𝐴 → 𝐵 = ∪ ( ( 𝑦 ∈ 𝐴 ↦ { 𝑦 } ) “ 𝐵 ) ) |
7 | 1 4 | eqtri | ⊢ 𝐹 = ( 𝑦 ∈ 𝐴 ↦ { 𝑦 } ) |
8 | 7 | imaeq1i | ⊢ ( 𝐹 “ 𝐵 ) = ( ( 𝑦 ∈ 𝐴 ↦ { 𝑦 } ) “ 𝐵 ) |
9 | 8 | unieqi | ⊢ ∪ ( 𝐹 “ 𝐵 ) = ∪ ( ( 𝑦 ∈ 𝐴 ↦ { 𝑦 } ) “ 𝐵 ) |
10 | 6 9 | eqtr4di | ⊢ ( 𝐵 ⊆ 𝐴 → 𝐵 = ∪ ( 𝐹 “ 𝐵 ) ) |