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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 2175 | . . . 4 ⊢ ℲyA | |
2 | nfab1 2177 | . . . 4 ⊢ Ⅎy{y ∣ φ} | |
3 | 1, 2 | nfiunxy 3674 | . . 3 ⊢ Ⅎy∪ x ∈ A {y ∣ φ} |
4 | nfab1 2177 | . . 3 ⊢ Ⅎy{y ∣ ∃x ∈ A φ} | |
5 | 3, 4 | cleqf 2198 | . 2 ⊢ (∪ x ∈ A {y ∣ φ} = {y ∣ ∃x ∈ A φ} ↔ ∀y(y ∈ ∪ x ∈ A {y ∣ φ} ↔ y ∈ {y ∣ ∃x ∈ A φ})) |
6 | abid 2025 | . . . 4 ⊢ (y ∈ {y ∣ φ} ↔ φ) | |
7 | 6 | rexbii 2325 | . . 3 ⊢ (∃x ∈ A y ∈ {y ∣ φ} ↔ ∃x ∈ A φ) |
8 | eliun 3652 | . . 3 ⊢ (y ∈ ∪ x ∈ A {y ∣ φ} ↔ ∃x ∈ A y ∈ {y ∣ φ}) | |
9 | abid 2025 | . . 3 ⊢ (y ∈ {y ∣ ∃x ∈ A φ} ↔ ∃x ∈ A φ) | |
10 | 7, 8, 9 | 3bitr4i 201 | . 2 ⊢ (y ∈ ∪ x ∈ A {y ∣ φ} ↔ y ∈ {y ∣ ∃x ∈ A φ}) |
11 | 5, 10 | mpgbir 1339 | 1 ⊢ ∪ x ∈ A {y ∣ φ} = {y ∣ ∃x ∈ A φ} |
Colors of variables: wff set class |
Syntax hints: ↔ wb 98 = wceq 1242 ∈ wcel 1390 {cab 2023 ∃wrex 2301 ∪ ciun 3648 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 |
This theorem depends on definitions: df-bi 110 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-rex 2306 df-v 2553 df-iun 3650 |
This theorem is referenced by: iunrab 3695 iunid 3703 dfimafn2 5166 |
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