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Mirrors > Home > ILE Home > Th. List > djudm | GIF version |
Description: The domain of the "domain-disjoint-union" is the disjoint union of the domains. Remark: its range is the (standard) union of the ranges. (Contributed by BJ, 10-Jul-2022.) |
Ref | Expression |
---|---|
djudm | ⊢ dom (𝐹 ⊔d 𝐺) = (dom 𝐹 ⊔ dom 𝐺) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-djud 7101 | . . 3 ⊢ (𝐹 ⊔d 𝐺) = ((𝐹 ∘ ◡(inl ↾ dom 𝐹)) ∪ (𝐺 ∘ ◡(inr ↾ dom 𝐺))) | |
2 | 1 | dmeqi 4828 | . 2 ⊢ dom (𝐹 ⊔d 𝐺) = dom ((𝐹 ∘ ◡(inl ↾ dom 𝐹)) ∪ (𝐺 ∘ ◡(inr ↾ dom 𝐺))) |
3 | dmun 4834 | . 2 ⊢ dom ((𝐹 ∘ ◡(inl ↾ dom 𝐹)) ∪ (𝐺 ∘ ◡(inr ↾ dom 𝐺))) = (dom (𝐹 ∘ ◡(inl ↾ dom 𝐹)) ∪ dom (𝐺 ∘ ◡(inr ↾ dom 𝐺))) | |
4 | dmco 5137 | . . . . 5 ⊢ dom (𝐹 ∘ ◡(inl ↾ dom 𝐹)) = (◡◡(inl ↾ dom 𝐹) “ dom 𝐹) | |
5 | imacnvcnv 5093 | . . . . 5 ⊢ (◡◡(inl ↾ dom 𝐹) “ dom 𝐹) = ((inl ↾ dom 𝐹) “ dom 𝐹) | |
6 | resima 4940 | . . . . . 6 ⊢ ((inl ↾ dom 𝐹) “ dom 𝐹) = (inl “ dom 𝐹) | |
7 | df-ima 4639 | . . . . . 6 ⊢ (inl “ dom 𝐹) = ran (inl ↾ dom 𝐹) | |
8 | 6, 7 | eqtri 2198 | . . . . 5 ⊢ ((inl ↾ dom 𝐹) “ dom 𝐹) = ran (inl ↾ dom 𝐹) |
9 | 4, 5, 8 | 3eqtri 2202 | . . . 4 ⊢ dom (𝐹 ∘ ◡(inl ↾ dom 𝐹)) = ran (inl ↾ dom 𝐹) |
10 | dmco 5137 | . . . . 5 ⊢ dom (𝐺 ∘ ◡(inr ↾ dom 𝐺)) = (◡◡(inr ↾ dom 𝐺) “ dom 𝐺) | |
11 | imacnvcnv 5093 | . . . . 5 ⊢ (◡◡(inr ↾ dom 𝐺) “ dom 𝐺) = ((inr ↾ dom 𝐺) “ dom 𝐺) | |
12 | resima 4940 | . . . . . 6 ⊢ ((inr ↾ dom 𝐺) “ dom 𝐺) = (inr “ dom 𝐺) | |
13 | df-ima 4639 | . . . . . 6 ⊢ (inr “ dom 𝐺) = ran (inr ↾ dom 𝐺) | |
14 | 12, 13 | eqtri 2198 | . . . . 5 ⊢ ((inr ↾ dom 𝐺) “ dom 𝐺) = ran (inr ↾ dom 𝐺) |
15 | 10, 11, 14 | 3eqtri 2202 | . . . 4 ⊢ dom (𝐺 ∘ ◡(inr ↾ dom 𝐺)) = ran (inr ↾ dom 𝐺) |
16 | 9, 15 | uneq12i 3287 | . . 3 ⊢ (dom (𝐹 ∘ ◡(inl ↾ dom 𝐹)) ∪ dom (𝐺 ∘ ◡(inr ↾ dom 𝐺))) = (ran (inl ↾ dom 𝐹) ∪ ran (inr ↾ dom 𝐺)) |
17 | djuunr 7064 | . . 3 ⊢ (ran (inl ↾ dom 𝐹) ∪ ran (inr ↾ dom 𝐺)) = (dom 𝐹 ⊔ dom 𝐺) | |
18 | 16, 17 | eqtri 2198 | . 2 ⊢ (dom (𝐹 ∘ ◡(inl ↾ dom 𝐹)) ∪ dom (𝐺 ∘ ◡(inr ↾ dom 𝐺))) = (dom 𝐹 ⊔ dom 𝐺) |
19 | 2, 3, 18 | 3eqtri 2202 | 1 ⊢ dom (𝐹 ⊔d 𝐺) = (dom 𝐹 ⊔ dom 𝐺) |
Colors of variables: wff set class |
Syntax hints: = wceq 1353 ∪ cun 3127 ◡ccnv 4625 dom cdm 4626 ran crn 4627 ↾ cres 4628 “ cima 4629 ∘ ccom 4630 ⊔ cdju 7035 inlcinl 7043 inrcinr 7044 ⊔d cdjud 7100 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4121 ax-nul 4129 ax-pow 4174 ax-pr 4209 ax-un 4433 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2739 df-sbc 2963 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-br 4004 df-opab 4065 df-mpt 4066 df-tr 4102 df-id 4293 df-iord 4366 df-on 4368 df-suc 4371 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-rn 4637 df-res 4638 df-ima 4639 df-iota 5178 df-fun 5218 df-fn 5219 df-f 5220 df-f1 5221 df-fo 5222 df-f1o 5223 df-fv 5224 df-1st 6140 df-2nd 6141 df-1o 6416 df-dju 7036 df-inl 7045 df-inr 7046 df-djud 7101 |
This theorem is referenced by: (None) |
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