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Mirrors > Home > MPE Home > Th. List > iinin2 | Structured version Visualization version GIF version |
Description: Indexed intersection of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use intiin 4974 to recover Enderton's theorem. (Contributed by Mario Carneiro, 19-Mar-2015.) |
Ref | Expression |
---|---|
iinin2 | ⊢ (𝐴 ≠ ∅ → ∩ 𝑥 ∈ 𝐴 (𝐵 ∩ 𝐶) = (𝐵 ∩ ∩ 𝑥 ∈ 𝐴 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | r19.28zv 4442 | . . . 4 ⊢ (𝐴 ≠ ∅ → (∀𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶) ↔ (𝑦 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐶))) | |
2 | elin 4166 | . . . . 5 ⊢ (𝑦 ∈ (𝐵 ∩ 𝐶) ↔ (𝑦 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) | |
3 | 2 | ralbii 3162 | . . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝑦 ∈ (𝐵 ∩ 𝐶) ↔ ∀𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) |
4 | eliin 4915 | . . . . . 6 ⊢ (𝑦 ∈ V → (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐶 ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐶)) | |
5 | 4 | elv 3497 | . . . . 5 ⊢ (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐶 ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐶) |
6 | 5 | anbi2i 622 | . . . 4 ⊢ ((𝑦 ∈ 𝐵 ∧ 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐶) ↔ (𝑦 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐶)) |
7 | 1, 3, 6 | 3bitr4g 315 | . . 3 ⊢ (𝐴 ≠ ∅ → (∀𝑥 ∈ 𝐴 𝑦 ∈ (𝐵 ∩ 𝐶) ↔ (𝑦 ∈ 𝐵 ∧ 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐶))) |
8 | eliin 4915 | . . . 4 ⊢ (𝑦 ∈ V → (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 (𝐵 ∩ 𝐶) ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ (𝐵 ∩ 𝐶))) | |
9 | 8 | elv 3497 | . . 3 ⊢ (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 (𝐵 ∩ 𝐶) ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ (𝐵 ∩ 𝐶)) |
10 | elin 4166 | . . 3 ⊢ (𝑦 ∈ (𝐵 ∩ ∩ 𝑥 ∈ 𝐴 𝐶) ↔ (𝑦 ∈ 𝐵 ∧ 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐶)) | |
11 | 7, 9, 10 | 3bitr4g 315 | . 2 ⊢ (𝐴 ≠ ∅ → (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 (𝐵 ∩ 𝐶) ↔ 𝑦 ∈ (𝐵 ∩ ∩ 𝑥 ∈ 𝐴 𝐶))) |
12 | 11 | eqrdv 2816 | 1 ⊢ (𝐴 ≠ ∅ → ∩ 𝑥 ∈ 𝐴 (𝐵 ∩ 𝐶) = (𝐵 ∩ ∩ 𝑥 ∈ 𝐴 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ≠ wne 3013 ∀wral 3135 Vcvv 3492 ∩ cin 3932 ∅c0 4288 ∩ ciin 4911 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-v 3494 df-dif 3936 df-in 3940 df-nul 4289 df-iin 4913 |
This theorem is referenced by: iinin1 4992 iniin2 41269 |
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