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Mirrors > Home > ILE Home > Th. List > iunab | GIF version |
Description: The indexed union of a class abstraction. (Contributed by NM, 27-Dec-2004.) |
Ref | Expression |
---|---|
iunab | ⊢ ∪ 𝑥 ∈ 𝐴 {𝑦 ∣ 𝜑} = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcv 2281 | . . . 4 ⊢ Ⅎ𝑦𝐴 | |
2 | nfab1 2283 | . . . 4 ⊢ Ⅎ𝑦{𝑦 ∣ 𝜑} | |
3 | 1, 2 | nfiunxy 3839 | . . 3 ⊢ Ⅎ𝑦∪ 𝑥 ∈ 𝐴 {𝑦 ∣ 𝜑} |
4 | nfab1 2283 | . . 3 ⊢ Ⅎ𝑦{𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑} | |
5 | 3, 4 | cleqf 2305 | . 2 ⊢ (∪ 𝑥 ∈ 𝐴 {𝑦 ∣ 𝜑} = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑} ↔ ∀𝑦(𝑦 ∈ ∪ 𝑥 ∈ 𝐴 {𝑦 ∣ 𝜑} ↔ 𝑦 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑})) |
6 | abid 2127 | . . . 4 ⊢ (𝑦 ∈ {𝑦 ∣ 𝜑} ↔ 𝜑) | |
7 | 6 | rexbii 2442 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 𝑦 ∈ {𝑦 ∣ 𝜑} ↔ ∃𝑥 ∈ 𝐴 𝜑) |
8 | eliun 3817 | . . 3 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 {𝑦 ∣ 𝜑} ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ {𝑦 ∣ 𝜑}) | |
9 | abid 2127 | . . 3 ⊢ (𝑦 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑} ↔ ∃𝑥 ∈ 𝐴 𝜑) | |
10 | 7, 8, 9 | 3bitr4i 211 | . 2 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 {𝑦 ∣ 𝜑} ↔ 𝑦 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑}) |
11 | 5, 10 | mpgbir 1429 | 1 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑦 ∣ 𝜑} = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑} |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 = wceq 1331 ∈ wcel 1480 {cab 2125 ∃wrex 2417 ∪ ciun 3813 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-iun 3815 |
This theorem is referenced by: iunrab 3860 iunid 3868 dfimafn2 5471 |
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