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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 4255 | . . . 4 ⊢ ((𝐴 ∪ 𝐵) ∩ I ) = ((𝐴 ∩ I ) ∪ (𝐵 ∩ I )) | |
2 | 1 | dmeqi 5776 | . . 3 ⊢ dom ((𝐴 ∪ 𝐵) ∩ I ) = dom ((𝐴 ∩ I ) ∪ (𝐵 ∩ I )) |
3 | dmun 5782 | . . 3 ⊢ dom ((𝐴 ∩ I ) ∪ (𝐵 ∩ I )) = (dom (𝐴 ∩ I ) ∪ dom (𝐵 ∩ I )) | |
4 | 2, 3 | eqtri 2847 | . 2 ⊢ dom ((𝐴 ∪ 𝐵) ∩ I ) = (dom (𝐴 ∩ I ) ∪ dom (𝐵 ∩ I )) |
5 | df-fix 33324 | . 2 ⊢ Fix (𝐴 ∪ 𝐵) = dom ((𝐴 ∪ 𝐵) ∩ I ) | |
6 | df-fix 33324 | . . 3 ⊢ Fix 𝐴 = dom (𝐴 ∩ I ) | |
7 | df-fix 33324 | . . 3 ⊢ Fix 𝐵 = dom (𝐵 ∩ I ) | |
8 | 6, 7 | uneq12i 4140 | . 2 ⊢ ( Fix 𝐴 ∪ Fix 𝐵) = (dom (𝐴 ∩ I ) ∪ dom (𝐵 ∩ I )) |
9 | 4, 5, 8 | 3eqtr4i 2857 | 1 ⊢ Fix (𝐴 ∪ 𝐵) = ( Fix 𝐴 ∪ Fix 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1536 ∪ cun 3937 ∩ cin 3938 I cid 5462 dom cdm 5558 Fix cfix 33300 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-rab 3150 df-v 3499 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 df-br 5070 df-dm 5568 df-fix 33324 |
This theorem is referenced by: (None) |
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