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Mirrors > Home > MPE Home > Th. List > Mathboxes > mptsnun | Structured version Visualization version GIF version |
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 |
---|---|
mptsnun.f | ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ {𝑥}) |
mptsnun.r | ⊢ 𝑅 = {𝑢 ∣ ∃𝑥 ∈ 𝐴 𝑢 = {𝑥}} |
Ref | Expression |
---|---|
mptsnun | ⊢ (𝐵 ⊆ 𝐴 → 𝐵 = ∪ (𝐹 “ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sneq 4579 | . . . . 5 ⊢ (𝑥 = 𝑦 → {𝑥} = {𝑦}) | |
2 | 1 | cbvmptv 5171 | . . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ {𝑥}) = (𝑦 ∈ 𝐴 ↦ {𝑦}) |
3 | 2 | eqcomi 2832 | . . 3 ⊢ (𝑦 ∈ 𝐴 ↦ {𝑦}) = (𝑥 ∈ 𝐴 ↦ {𝑥}) |
4 | mptsnun.r | . . 3 ⊢ 𝑅 = {𝑢 ∣ ∃𝑥 ∈ 𝐴 𝑢 = {𝑥}} | |
5 | 3, 4 | mptsnunlem 34621 | . 2 ⊢ (𝐵 ⊆ 𝐴 → 𝐵 = ∪ ((𝑦 ∈ 𝐴 ↦ {𝑦}) “ 𝐵)) |
6 | mptsnun.f | . . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ {𝑥}) | |
7 | 6, 2 | eqtri 2846 | . . . 4 ⊢ 𝐹 = (𝑦 ∈ 𝐴 ↦ {𝑦}) |
8 | 7 | imaeq1i 5928 | . . 3 ⊢ (𝐹 “ 𝐵) = ((𝑦 ∈ 𝐴 ↦ {𝑦}) “ 𝐵) |
9 | 8 | unieqi 4853 | . 2 ⊢ ∪ (𝐹 “ 𝐵) = ∪ ((𝑦 ∈ 𝐴 ↦ {𝑦}) “ 𝐵) |
10 | 5, 9 | syl6eqr 2876 | 1 ⊢ (𝐵 ⊆ 𝐴 → 𝐵 = ∪ (𝐹 “ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 {cab 2801 ∃wrex 3141 ⊆ wss 3938 {csn 4569 ∪ cuni 4840 ↦ cmpt 5148 “ cima 5560 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-xp 5563 df-rel 5564 df-cnv 5565 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 |
This theorem is referenced by: dissneqlem 34623 |
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