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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 3875 | . . . 4 ⊢ ((𝐴 ∪ 𝐵) ∩ I ) = ((𝐴 ∩ I ) ∪ (𝐵 ∩ I )) | |
2 | 1 | dmeqi 5325 | . . 3 ⊢ dom ((𝐴 ∪ 𝐵) ∩ I ) = dom ((𝐴 ∩ I ) ∪ (𝐵 ∩ I )) |
3 | dmun 5331 | . . 3 ⊢ dom ((𝐴 ∩ I ) ∪ (𝐵 ∩ I )) = (dom (𝐴 ∩ I ) ∪ dom (𝐵 ∩ I )) | |
4 | 2, 3 | eqtri 2644 | . 2 ⊢ dom ((𝐴 ∪ 𝐵) ∩ I ) = (dom (𝐴 ∩ I ) ∪ dom (𝐵 ∩ I )) |
5 | df-fix 31966 | . 2 ⊢ Fix (𝐴 ∪ 𝐵) = dom ((𝐴 ∪ 𝐵) ∩ I ) | |
6 | df-fix 31966 | . . 3 ⊢ Fix 𝐴 = dom (𝐴 ∩ I ) | |
7 | df-fix 31966 | . . 3 ⊢ Fix 𝐵 = dom (𝐵 ∩ I ) | |
8 | 6, 7 | uneq12i 3765 | . 2 ⊢ ( Fix 𝐴 ∪ Fix 𝐵) = (dom (𝐴 ∩ I ) ∪ dom (𝐵 ∩ I )) |
9 | 4, 5, 8 | 3eqtr4i 2654 | 1 ⊢ Fix (𝐴 ∪ 𝐵) = ( Fix 𝐴 ∪ Fix 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1483 ∪ cun 3572 ∩ cin 3573 I cid 5023 dom cdm 5114 Fix cfix 31942 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-br 4654 df-dm 5124 df-fix 31966 |
This theorem is referenced by: (None) |
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