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Mirrors > Home > MPE Home > Th. List > viin | Structured version Visualization version GIF version |
Description: Indexed intersection with a universal index class. When 𝐴 doesn't depend on 𝑥, this evaluates to 𝐴 by 19.3 2188 and abid2 2866. When 𝐴 = 𝑥, this evaluates to ∅ by intiin 5056 and intv 5358. (Contributed by NM, 11-Sep-2008.) |
Ref | Expression |
---|---|
viin | ⊢ ∩ 𝑥 ∈ V 𝐴 = {𝑦 ∣ ∀𝑥 𝑦 ∈ 𝐴} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-iin 4994 | . 2 ⊢ ∩ 𝑥 ∈ V 𝐴 = {𝑦 ∣ ∀𝑥 ∈ V 𝑦 ∈ 𝐴} | |
2 | ralv 3494 | . . 3 ⊢ (∀𝑥 ∈ V 𝑦 ∈ 𝐴 ↔ ∀𝑥 𝑦 ∈ 𝐴) | |
3 | 2 | abbii 2797 | . 2 ⊢ {𝑦 ∣ ∀𝑥 ∈ V 𝑦 ∈ 𝐴} = {𝑦 ∣ ∀𝑥 𝑦 ∈ 𝐴} |
4 | 1, 3 | eqtri 2755 | 1 ⊢ ∩ 𝑥 ∈ V 𝐴 = {𝑦 ∣ ∀𝑥 𝑦 ∈ 𝐴} |
Colors of variables: wff setvar class |
Syntax hints: ∀wal 1532 = wceq 1534 ∈ wcel 2099 {cab 2704 ∀wral 3056 Vcvv 3469 ∩ ciin 4992 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2698 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1537 df-ex 1775 df-sb 2061 df-clab 2705 df-cleq 2719 df-clel 2805 df-ral 3057 df-v 3471 df-iin 4994 |
This theorem is referenced by: (None) |
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