![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > iunin2 | Structured version Visualization version GIF version |
Description: Indexed union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use uniiun 4887 to recover Enderton's theorem. (Contributed by NM, 26-Mar-2004.) |
Ref | Expression |
---|---|
iunin2 | ⊢ ∪ 𝑥 ∈ 𝐴 (𝐵 ∩ 𝐶) = (𝐵 ∩ ∪ 𝑥 ∈ 𝐴 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | r19.42v 3313 | . . . 4 ⊢ (∃𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶) ↔ (𝑦 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐶)) | |
2 | elin 4096 | . . . . 5 ⊢ (𝑦 ∈ (𝐵 ∩ 𝐶) ↔ (𝑦 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) | |
3 | 2 | rexbii 3213 | . . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝑦 ∈ (𝐵 ∩ 𝐶) ↔ ∃𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) |
4 | eliun 4835 | . . . . 5 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐶 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐶) | |
5 | 4 | anbi2i 622 | . . . 4 ⊢ ((𝑦 ∈ 𝐵 ∧ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐶) ↔ (𝑦 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐶)) |
6 | 1, 3, 5 | 3bitr4i 304 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 𝑦 ∈ (𝐵 ∩ 𝐶) ↔ (𝑦 ∈ 𝐵 ∧ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐶)) |
7 | eliun 4835 | . . 3 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 (𝐵 ∩ 𝐶) ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ (𝐵 ∩ 𝐶)) | |
8 | elin 4096 | . . 3 ⊢ (𝑦 ∈ (𝐵 ∩ ∪ 𝑥 ∈ 𝐴 𝐶) ↔ (𝑦 ∈ 𝐵 ∧ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐶)) | |
9 | 6, 7, 8 | 3bitr4i 304 | . 2 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 (𝐵 ∩ 𝐶) ↔ 𝑦 ∈ (𝐵 ∩ ∪ 𝑥 ∈ 𝐴 𝐶)) |
10 | 9 | eqriv 2794 | 1 ⊢ ∪ 𝑥 ∈ 𝐴 (𝐵 ∩ 𝐶) = (𝐵 ∩ ∪ 𝑥 ∈ 𝐴 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 = wceq 1525 ∈ wcel 2083 ∃wrex 3108 ∩ cin 3864 ∪ ciun 4831 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-8 2085 ax-9 2093 ax-10 2114 ax-11 2128 ax-12 2143 ax-ext 2771 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-tru 1528 df-ex 1766 df-nf 1770 df-sb 2045 df-clab 2778 df-cleq 2790 df-clel 2865 df-nfc 2937 df-ral 3112 df-rex 3113 df-v 3442 df-in 3872 df-iun 4833 |
This theorem is referenced by: iunin1 4899 2iunin 4903 resiun2 5762 infssuni 8668 kmlem11 9439 cmpsublem 21695 cmpsub 21696 kgentopon 21834 metnrmlem3 23156 ovoliunlem1 23790 voliunlem1 23838 voliunlem2 23839 uniioombllem2 23871 uniioombllem4 23874 volsup2 23893 itg1addlem5 23988 itg1climres 24002 uniin2 29989 carsgclctunlem2 31190 cvmscld 32130 cnambfre 34492 ftc1anclem6 34524 heiborlem3 34644 carageniuncllem2 42368 |
Copyright terms: Public domain | W3C validator |