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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 4535 | . . . . 5 ⊢ (𝑥 = 𝑦 → {𝑥} = {𝑦}) | |
2 | 1 | cbvmptv 5133 | . . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ {𝑥}) = (𝑦 ∈ 𝐴 ↦ {𝑦}) |
3 | 2 | eqcomi 2807 | . . 3 ⊢ (𝑦 ∈ 𝐴 ↦ {𝑦}) = (𝑥 ∈ 𝐴 ↦ {𝑥}) |
4 | mptsnun.r | . . 3 ⊢ 𝑅 = {𝑢 ∣ ∃𝑥 ∈ 𝐴 𝑢 = {𝑥}} | |
5 | 3, 4 | mptsnunlem 34755 | . 2 ⊢ (𝐵 ⊆ 𝐴 → 𝐵 = ∪ ((𝑦 ∈ 𝐴 ↦ {𝑦}) “ 𝐵)) |
6 | mptsnun.f | . . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ {𝑥}) | |
7 | 6, 2 | eqtri 2821 | . . . 4 ⊢ 𝐹 = (𝑦 ∈ 𝐴 ↦ {𝑦}) |
8 | 7 | imaeq1i 5893 | . . 3 ⊢ (𝐹 “ 𝐵) = ((𝑦 ∈ 𝐴 ↦ {𝑦}) “ 𝐵) |
9 | 8 | unieqi 4813 | . 2 ⊢ ∪ (𝐹 “ 𝐵) = ∪ ((𝑦 ∈ 𝐴 ↦ {𝑦}) “ 𝐵) |
10 | 5, 9 | eqtr4di 2851 | 1 ⊢ (𝐵 ⊆ 𝐴 → 𝐵 = ∪ (𝐹 “ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 {cab 2776 ∃wrex 3107 ⊆ wss 3881 {csn 4525 ∪ cuni 4800 ↦ cmpt 5110 “ cima 5522 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-xp 5525 df-rel 5526 df-cnv 5527 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 |
This theorem is referenced by: dissneqlem 34757 |
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