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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 4408 | . . . . 5 ⊢ (𝑥 = 𝑦 → {𝑥} = {𝑦}) | |
2 | 1 | cbvmptv 4987 | . . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ {𝑥}) = (𝑦 ∈ 𝐴 ↦ {𝑦}) |
3 | 2 | eqcomi 2787 | . . 3 ⊢ (𝑦 ∈ 𝐴 ↦ {𝑦}) = (𝑥 ∈ 𝐴 ↦ {𝑥}) |
4 | mptsnun.r | . . 3 ⊢ 𝑅 = {𝑢 ∣ ∃𝑥 ∈ 𝐴 𝑢 = {𝑥}} | |
5 | 3, 4 | mptsnunlem 33784 | . 2 ⊢ (𝐵 ⊆ 𝐴 → 𝐵 = ∪ ((𝑦 ∈ 𝐴 ↦ {𝑦}) “ 𝐵)) |
6 | mptsnun.f | . . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ {𝑥}) | |
7 | 6, 2 | eqtri 2802 | . . . 4 ⊢ 𝐹 = (𝑦 ∈ 𝐴 ↦ {𝑦}) |
8 | 7 | imaeq1i 5719 | . . 3 ⊢ (𝐹 “ 𝐵) = ((𝑦 ∈ 𝐴 ↦ {𝑦}) “ 𝐵) |
9 | 8 | unieqi 4682 | . 2 ⊢ ∪ (𝐹 “ 𝐵) = ∪ ((𝑦 ∈ 𝐴 ↦ {𝑦}) “ 𝐵) |
10 | 5, 9 | syl6eqr 2832 | 1 ⊢ (𝐵 ⊆ 𝐴 → 𝐵 = ∪ (𝐹 “ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1601 {cab 2763 ∃wrex 3091 ⊆ wss 3792 {csn 4398 ∪ cuni 4673 ↦ cmpt 4967 “ cima 5360 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5019 ax-nul 5027 ax-pr 5140 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3400 df-sbc 3653 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4674 df-br 4889 df-opab 4951 df-mpt 4968 df-xp 5363 df-rel 5364 df-cnv 5365 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 |
This theorem is referenced by: dissneqlem 33786 |
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