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Mirrors > Home > MPE Home > Th. List > Mathboxes > fixun | Structured version Visualization version GIF version |
Description: The fixpoint operator distributes over union. (Contributed by Scott Fenton, 16-Apr-2012.) |
Ref | Expression |
---|---|
fixun | ⊢ Fix (𝐴 ∪ 𝐵) = ( Fix 𝐴 ∪ Fix 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | indir 3834 | . . . 4 ⊢ ((𝐴 ∪ 𝐵) ∩ I ) = ((𝐴 ∩ I ) ∪ (𝐵 ∩ I )) | |
2 | 1 | dmeqi 5247 | . . 3 ⊢ dom ((𝐴 ∪ 𝐵) ∩ I ) = dom ((𝐴 ∩ I ) ∪ (𝐵 ∩ I )) |
3 | dmun 5253 | . . 3 ⊢ dom ((𝐴 ∩ I ) ∪ (𝐵 ∩ I )) = (dom (𝐴 ∩ I ) ∪ dom (𝐵 ∩ I )) | |
4 | 2, 3 | eqtri 2632 | . 2 ⊢ dom ((𝐴 ∪ 𝐵) ∩ I ) = (dom (𝐴 ∩ I ) ∪ dom (𝐵 ∩ I )) |
5 | df-fix 31135 | . 2 ⊢ Fix (𝐴 ∪ 𝐵) = dom ((𝐴 ∪ 𝐵) ∩ I ) | |
6 | df-fix 31135 | . . 3 ⊢ Fix 𝐴 = dom (𝐴 ∩ I ) | |
7 | df-fix 31135 | . . 3 ⊢ Fix 𝐵 = dom (𝐵 ∩ I ) | |
8 | 6, 7 | uneq12i 3727 | . 2 ⊢ ( Fix 𝐴 ∪ Fix 𝐵) = (dom (𝐴 ∩ I ) ∪ dom (𝐵 ∩ I )) |
9 | 4, 5, 8 | 3eqtr4i 2642 | 1 ⊢ Fix (𝐴 ∪ 𝐵) = ( Fix 𝐴 ∪ Fix 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 ∪ cun 3538 ∩ cin 3539 I cid 4948 dom cdm 5038 Fix cfix 31111 |
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-3an 1033 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-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-dm 5048 df-fix 31135 |
This theorem is referenced by: (None) |
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