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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 2202 and abid2 2959. When 𝐴 = 𝑥, this evaluates to ∅ by intiin 4985 and intv 5266. (Contributed by NM, 11-Sep-2008.) |
Ref | Expression |
---|---|
viin | ⊢ ∩ 𝑥 ∈ V 𝐴 = {𝑦 ∣ ∀𝑥 𝑦 ∈ 𝐴} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-iin 4924 | . 2 ⊢ ∩ 𝑥 ∈ V 𝐴 = {𝑦 ∣ ∀𝑥 ∈ V 𝑦 ∈ 𝐴} | |
2 | ralv 3521 | . . 3 ⊢ (∀𝑥 ∈ V 𝑦 ∈ 𝐴 ↔ ∀𝑥 𝑦 ∈ 𝐴) | |
3 | 2 | abbii 2888 | . 2 ⊢ {𝑦 ∣ ∀𝑥 ∈ V 𝑦 ∈ 𝐴} = {𝑦 ∣ ∀𝑥 𝑦 ∈ 𝐴} |
4 | 1, 3 | eqtri 2846 | 1 ⊢ ∩ 𝑥 ∈ V 𝐴 = {𝑦 ∣ ∀𝑥 𝑦 ∈ 𝐴} |
Colors of variables: wff setvar class |
Syntax hints: ∀wal 1535 = wceq 1537 ∈ wcel 2114 {cab 2801 ∀wral 3140 Vcvv 3496 ∩ ciin 4922 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1781 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-ral 3145 df-v 3498 df-iin 4924 |
This theorem is referenced by: (None) |
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