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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 3412 | . . . . 5 ⊢ {𝑥} = {𝑦 ∣ 𝑦 = 𝑥} | |
2 | equcom 1634 | . . . . . 6 ⊢ (𝑦 = 𝑥 ↔ 𝑥 = 𝑦) | |
3 | 2 | abbii 2195 | . . . . 5 ⊢ {𝑦 ∣ 𝑦 = 𝑥} = {𝑦 ∣ 𝑥 = 𝑦} |
4 | 1, 3 | eqtri 2102 | . . . 4 ⊢ {𝑥} = {𝑦 ∣ 𝑥 = 𝑦} |
5 | 4 | a1i 9 | . . 3 ⊢ (𝑥 ∈ 𝐴 → {𝑥} = {𝑦 ∣ 𝑥 = 𝑦}) |
6 | 5 | iuneq2i 3704 | . 2 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑥} = ∪ 𝑥 ∈ 𝐴 {𝑦 ∣ 𝑥 = 𝑦} |
7 | iunab 3732 | . . 3 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑦 ∣ 𝑥 = 𝑦} = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑥 = 𝑦} | |
8 | risset 2395 | . . . 4 ⊢ (𝑦 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 𝑥 = 𝑦) | |
9 | 8 | abbii 2195 | . . 3 ⊢ {𝑦 ∣ 𝑦 ∈ 𝐴} = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑥 = 𝑦} |
10 | abid2 2200 | . . 3 ⊢ {𝑦 ∣ 𝑦 ∈ 𝐴} = 𝐴 | |
11 | 7, 9, 10 | 3eqtr2i 2108 | . 2 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑦 ∣ 𝑥 = 𝑦} = 𝐴 |
12 | 6, 11 | eqtri 2102 | 1 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑥} = 𝐴 |
Colors of variables: wff set class |
Syntax hints: = wceq 1285 ∈ wcel 1434 {cab 2068 ∃wrex 2350 {csn 3406 ∪ ciun 3686 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2064 |
This theorem depends on definitions: df-bi 115 df-tru 1288 df-nf 1391 df-sb 1687 df-clab 2069 df-cleq 2075 df-clel 2078 df-nfc 2209 df-ral 2354 df-rex 2355 df-v 2604 df-in 2980 df-ss 2987 df-sn 3412 df-iun 3688 |
This theorem is referenced by: iunxpconst 4426 xpexgALT 5791 uniqs 6230 |
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