Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > iinuni | Structured version Visualization version GIF version |
Description: A relationship involving union and indexed intersection. Exercise 23 of [Enderton] p. 33. (Contributed by NM, 25-Nov-2003.) (Proof shortened by Mario Carneiro, 17-Nov-2016.) |
Ref | Expression |
---|---|
iinuni | ⊢ (𝐴 ∪ ∩ 𝐵) = ∩ 𝑥 ∈ 𝐵 (𝐴 ∪ 𝑥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | r19.32v 3064 | . . . 4 ⊢ (∀𝑥 ∈ 𝐵 (𝑦 ∈ 𝐴 ∨ 𝑦 ∈ 𝑥) ↔ (𝑦 ∈ 𝐴 ∨ ∀𝑥 ∈ 𝐵 𝑦 ∈ 𝑥)) | |
2 | elun 3715 | . . . . 5 ⊢ (𝑦 ∈ (𝐴 ∪ 𝑥) ↔ (𝑦 ∈ 𝐴 ∨ 𝑦 ∈ 𝑥)) | |
3 | 2 | ralbii 2963 | . . . 4 ⊢ (∀𝑥 ∈ 𝐵 𝑦 ∈ (𝐴 ∪ 𝑥) ↔ ∀𝑥 ∈ 𝐵 (𝑦 ∈ 𝐴 ∨ 𝑦 ∈ 𝑥)) |
4 | vex 3176 | . . . . . 6 ⊢ 𝑦 ∈ V | |
5 | 4 | elint2 4417 | . . . . 5 ⊢ (𝑦 ∈ ∩ 𝐵 ↔ ∀𝑥 ∈ 𝐵 𝑦 ∈ 𝑥) |
6 | 5 | orbi2i 540 | . . . 4 ⊢ ((𝑦 ∈ 𝐴 ∨ 𝑦 ∈ ∩ 𝐵) ↔ (𝑦 ∈ 𝐴 ∨ ∀𝑥 ∈ 𝐵 𝑦 ∈ 𝑥)) |
7 | 1, 3, 6 | 3bitr4ri 292 | . . 3 ⊢ ((𝑦 ∈ 𝐴 ∨ 𝑦 ∈ ∩ 𝐵) ↔ ∀𝑥 ∈ 𝐵 𝑦 ∈ (𝐴 ∪ 𝑥)) |
8 | 7 | abbii 2726 | . 2 ⊢ {𝑦 ∣ (𝑦 ∈ 𝐴 ∨ 𝑦 ∈ ∩ 𝐵)} = {𝑦 ∣ ∀𝑥 ∈ 𝐵 𝑦 ∈ (𝐴 ∪ 𝑥)} |
9 | df-un 3545 | . 2 ⊢ (𝐴 ∪ ∩ 𝐵) = {𝑦 ∣ (𝑦 ∈ 𝐴 ∨ 𝑦 ∈ ∩ 𝐵)} | |
10 | df-iin 4458 | . 2 ⊢ ∩ 𝑥 ∈ 𝐵 (𝐴 ∪ 𝑥) = {𝑦 ∣ ∀𝑥 ∈ 𝐵 𝑦 ∈ (𝐴 ∪ 𝑥)} | |
11 | 8, 9, 10 | 3eqtr4i 2642 | 1 ⊢ (𝐴 ∪ ∩ 𝐵) = ∩ 𝑥 ∈ 𝐵 (𝐴 ∪ 𝑥) |
Colors of variables: wff setvar class |
Syntax hints: ∨ wo 382 = wceq 1475 ∈ wcel 1977 {cab 2596 ∀wral 2896 ∪ cun 3538 ∩ cint 4410 ∩ ciin 4456 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-v 3175 df-un 3545 df-int 4411 df-iin 4458 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |