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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 4580 | . . . . 5 ⊢ (𝑥 = 𝑦 → {𝑥} = {𝑦}) | |
2 | 1 | cbvmptv 5172 | . . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ {𝑥}) = (𝑦 ∈ 𝐴 ↦ {𝑦}) |
3 | 2 | eqcomi 2833 | . . 3 ⊢ (𝑦 ∈ 𝐴 ↦ {𝑦}) = (𝑥 ∈ 𝐴 ↦ {𝑥}) |
4 | mptsnun.r | . . 3 ⊢ 𝑅 = {𝑢 ∣ ∃𝑥 ∈ 𝐴 𝑢 = {𝑥}} | |
5 | 3, 4 | mptsnunlem 34623 | . 2 ⊢ (𝐵 ⊆ 𝐴 → 𝐵 = ∪ ((𝑦 ∈ 𝐴 ↦ {𝑦}) “ 𝐵)) |
6 | mptsnun.f | . . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ {𝑥}) | |
7 | 6, 2 | eqtri 2847 | . . . 4 ⊢ 𝐹 = (𝑦 ∈ 𝐴 ↦ {𝑦}) |
8 | 7 | imaeq1i 5929 | . . 3 ⊢ (𝐹 “ 𝐵) = ((𝑦 ∈ 𝐴 ↦ {𝑦}) “ 𝐵) |
9 | 8 | unieqi 4854 | . 2 ⊢ ∪ (𝐹 “ 𝐵) = ∪ ((𝑦 ∈ 𝐴 ↦ {𝑦}) “ 𝐵) |
10 | 5, 9 | syl6eqr 2877 | 1 ⊢ (𝐵 ⊆ 𝐴 → 𝐵 = ∪ (𝐹 “ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 {cab 2802 ∃wrex 3142 ⊆ wss 3939 {csn 4570 ∪ cuni 4841 ↦ cmpt 5149 “ cima 5561 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pr 5333 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-sbc 3776 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-opab 5132 df-mpt 5150 df-xp 5564 df-rel 5565 df-cnv 5566 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 |
This theorem is referenced by: dissneqlem 34625 |
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