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Mirrors > Home > ILE Home > Th. List > iinab | GIF version |
Description: Indexed intersection of a class builder. (Contributed by NM, 6-Dec-2011.) |
Ref | Expression |
---|---|
iinab | ⊢ ∩ 𝑥 ∈ 𝐴 {𝑦 ∣ 𝜑} = {𝑦 ∣ ∀𝑥 ∈ 𝐴 𝜑} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcv 2178 | . . . 4 ⊢ Ⅎ𝑦𝐴 | |
2 | nfab1 2180 | . . . 4 ⊢ Ⅎ𝑦{𝑦 ∣ 𝜑} | |
3 | 1, 2 | nfiinxy 3684 | . . 3 ⊢ Ⅎ𝑦∩ 𝑥 ∈ 𝐴 {𝑦 ∣ 𝜑} |
4 | nfab1 2180 | . . 3 ⊢ Ⅎ𝑦{𝑦 ∣ ∀𝑥 ∈ 𝐴 𝜑} | |
5 | 3, 4 | cleqf 2201 | . 2 ⊢ (∩ 𝑥 ∈ 𝐴 {𝑦 ∣ 𝜑} = {𝑦 ∣ ∀𝑥 ∈ 𝐴 𝜑} ↔ ∀𝑦(𝑦 ∈ ∩ 𝑥 ∈ 𝐴 {𝑦 ∣ 𝜑} ↔ 𝑦 ∈ {𝑦 ∣ ∀𝑥 ∈ 𝐴 𝜑})) |
6 | abid 2028 | . . . 4 ⊢ (𝑦 ∈ {𝑦 ∣ 𝜑} ↔ 𝜑) | |
7 | 6 | ralbii 2330 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 𝑦 ∈ {𝑦 ∣ 𝜑} ↔ ∀𝑥 ∈ 𝐴 𝜑) |
8 | vex 2560 | . . . 4 ⊢ 𝑦 ∈ V | |
9 | eliin 3662 | . . . 4 ⊢ (𝑦 ∈ V → (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 {𝑦 ∣ 𝜑} ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ {𝑦 ∣ 𝜑})) | |
10 | 8, 9 | ax-mp 7 | . . 3 ⊢ (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 {𝑦 ∣ 𝜑} ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ {𝑦 ∣ 𝜑}) |
11 | abid 2028 | . . 3 ⊢ (𝑦 ∈ {𝑦 ∣ ∀𝑥 ∈ 𝐴 𝜑} ↔ ∀𝑥 ∈ 𝐴 𝜑) | |
12 | 7, 10, 11 | 3bitr4i 201 | . 2 ⊢ (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 {𝑦 ∣ 𝜑} ↔ 𝑦 ∈ {𝑦 ∣ ∀𝑥 ∈ 𝐴 𝜑}) |
13 | 5, 12 | mpgbir 1342 | 1 ⊢ ∩ 𝑥 ∈ 𝐴 {𝑦 ∣ 𝜑} = {𝑦 ∣ ∀𝑥 ∈ 𝐴 𝜑} |
Colors of variables: wff set class |
Syntax hints: ↔ wb 98 = wceq 1243 ∈ wcel 1393 {cab 2026 ∀wral 2306 Vcvv 2557 ∩ ciin 3658 |
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 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-v 2559 df-iin 3660 |
This theorem is referenced by: iinrabm 3719 |
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