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Mirrors > Home > ILE Home > Th. List > iunid | GIF version |
Description: An indexed union of singletons recovers the index set. (Contributed by NM, 6-Sep-2005.) |
Ref | Expression |
---|---|
iunid | ⊢ ∪ 𝑥 ∈ 𝐴 {𝑥} = 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-sn 3497 | . . . . 5 ⊢ {𝑥} = {𝑦 ∣ 𝑦 = 𝑥} | |
2 | equcom 1663 | . . . . . 6 ⊢ (𝑦 = 𝑥 ↔ 𝑥 = 𝑦) | |
3 | 2 | abbii 2228 | . . . . 5 ⊢ {𝑦 ∣ 𝑦 = 𝑥} = {𝑦 ∣ 𝑥 = 𝑦} |
4 | 1, 3 | eqtri 2133 | . . . 4 ⊢ {𝑥} = {𝑦 ∣ 𝑥 = 𝑦} |
5 | 4 | a1i 9 | . . 3 ⊢ (𝑥 ∈ 𝐴 → {𝑥} = {𝑦 ∣ 𝑥 = 𝑦}) |
6 | 5 | iuneq2i 3795 | . 2 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑥} = ∪ 𝑥 ∈ 𝐴 {𝑦 ∣ 𝑥 = 𝑦} |
7 | iunab 3823 | . . 3 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑦 ∣ 𝑥 = 𝑦} = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑥 = 𝑦} | |
8 | risset 2435 | . . . 4 ⊢ (𝑦 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 𝑥 = 𝑦) | |
9 | 8 | abbii 2228 | . . 3 ⊢ {𝑦 ∣ 𝑦 ∈ 𝐴} = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑥 = 𝑦} |
10 | abid2 2233 | . . 3 ⊢ {𝑦 ∣ 𝑦 ∈ 𝐴} = 𝐴 | |
11 | 7, 9, 10 | 3eqtr2i 2139 | . 2 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑦 ∣ 𝑥 = 𝑦} = 𝐴 |
12 | 6, 11 | eqtri 2133 | 1 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑥} = 𝐴 |
Colors of variables: wff set class |
Syntax hints: = wceq 1312 ∈ wcel 1461 {cab 2099 ∃wrex 2389 {csn 3491 ∪ ciun 3777 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 681 ax-5 1404 ax-7 1405 ax-gen 1406 ax-ie1 1450 ax-ie2 1451 ax-8 1463 ax-10 1464 ax-11 1465 ax-i12 1466 ax-bndl 1467 ax-4 1468 ax-17 1487 ax-i9 1491 ax-ial 1495 ax-i5r 1496 ax-ext 2095 |
This theorem depends on definitions: df-bi 116 df-tru 1315 df-nf 1418 df-sb 1717 df-clab 2100 df-cleq 2106 df-clel 2109 df-nfc 2242 df-ral 2393 df-rex 2394 df-v 2657 df-in 3041 df-ss 3048 df-sn 3497 df-iun 3779 |
This theorem is referenced by: abnexg 4325 iunxpconst 4557 xpexgALT 5983 uniqs 6439 |
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