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Mirrors > Home > NFE Home > Th. List > viin | GIF version |
Description: Indexed intersection with a universal index class. When A doesn't depend on x, this evaluates to A by 19.3 1785 and abid2 2470. When A = x, this evaluates to ∅ by intiin 4020 and intv in set.mm. (Contributed by NM, 11-Sep-2008.) |
Ref | Expression |
---|---|
viin | ⊢ ∩x ∈ V A = {y ∣ ∀x y ∈ A} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-iin 3972 | . 2 ⊢ ∩x ∈ V A = {y ∣ ∀x ∈ V y ∈ A} | |
2 | ralv 2872 | . . 3 ⊢ (∀x ∈ V y ∈ A ↔ ∀x y ∈ A) | |
3 | 2 | abbii 2465 | . 2 ⊢ {y ∣ ∀x ∈ V y ∈ A} = {y ∣ ∀x y ∈ A} |
4 | 1, 3 | eqtri 2373 | 1 ⊢ ∩x ∈ V A = {y ∣ ∀x y ∈ A} |
Colors of variables: wff setvar class |
Syntax hints: ∀wal 1540 = wceq 1642 ∈ wcel 1710 {cab 2339 ∀wral 2614 Vcvv 2859 ∩ciin 3970 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-an 360 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-ral 2619 df-v 2861 df-iin 3972 |
This theorem is referenced by: (None) |
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