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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 4633 | . . . . 5 ⊢ (𝑥 = 𝑦 → {𝑥} = {𝑦}) | |
2 | 1 | cbvmptv 5258 | . . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ {𝑥}) = (𝑦 ∈ 𝐴 ↦ {𝑦}) |
3 | 2 | eqcomi 2735 | . . 3 ⊢ (𝑦 ∈ 𝐴 ↦ {𝑦}) = (𝑥 ∈ 𝐴 ↦ {𝑥}) |
4 | mptsnun.r | . . 3 ⊢ 𝑅 = {𝑢 ∣ ∃𝑥 ∈ 𝐴 𝑢 = {𝑥}} | |
5 | 3, 4 | mptsnunlem 37058 | . 2 ⊢ (𝐵 ⊆ 𝐴 → 𝐵 = ∪ ((𝑦 ∈ 𝐴 ↦ {𝑦}) “ 𝐵)) |
6 | mptsnun.f | . . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ {𝑥}) | |
7 | 6, 2 | eqtri 2754 | . . . 4 ⊢ 𝐹 = (𝑦 ∈ 𝐴 ↦ {𝑦}) |
8 | 7 | imaeq1i 6058 | . . 3 ⊢ (𝐹 “ 𝐵) = ((𝑦 ∈ 𝐴 ↦ {𝑦}) “ 𝐵) |
9 | 8 | unieqi 4917 | . 2 ⊢ ∪ (𝐹 “ 𝐵) = ∪ ((𝑦 ∈ 𝐴 ↦ {𝑦}) “ 𝐵) |
10 | 5, 9 | eqtr4di 2784 | 1 ⊢ (𝐵 ⊆ 𝐴 → 𝐵 = ∪ (𝐹 “ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 {cab 2703 ∃wrex 3060 ⊆ wss 3946 {csn 4623 ∪ cuni 4905 ↦ cmpt 5228 “ cima 5677 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5296 ax-nul 5303 ax-pr 5425 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-sbc 3776 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4323 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-br 5146 df-opab 5208 df-mpt 5229 df-xp 5680 df-rel 5681 df-cnv 5682 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 |
This theorem is referenced by: dissneqlem 37060 |
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