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Mirrors > Home > ILE Home > Th. List > viin | GIF version |
Description: Indexed intersection with a universal index class. (Contributed by NM, 11-Sep-2008.) |
Ref | Expression |
---|---|
viin | ⊢ ∩ 𝑥 ∈ V 𝐴 = {𝑦 ∣ ∀𝑥 𝑦 ∈ 𝐴} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-iin 3660 | . 2 ⊢ ∩ 𝑥 ∈ V 𝐴 = {𝑦 ∣ ∀𝑥 ∈ V 𝑦 ∈ 𝐴} | |
2 | ralv 2571 | . . 3 ⊢ (∀𝑥 ∈ V 𝑦 ∈ 𝐴 ↔ ∀𝑥 𝑦 ∈ 𝐴) | |
3 | 2 | abbii 2153 | . 2 ⊢ {𝑦 ∣ ∀𝑥 ∈ V 𝑦 ∈ 𝐴} = {𝑦 ∣ ∀𝑥 𝑦 ∈ 𝐴} |
4 | 1, 3 | eqtri 2060 | 1 ⊢ ∩ 𝑥 ∈ V 𝐴 = {𝑦 ∣ ∀𝑥 𝑦 ∈ 𝐴} |
Colors of variables: wff set class |
Syntax hints: ∀wal 1241 = 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-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-11 1397 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-ral 2311 df-v 2559 df-iin 3660 |
This theorem is referenced by: (None) |
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