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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 7080 | . . 3 ⊢ (𝐹 ⊔d 𝐺) = ((𝐹 ∘ ◡(inl ↾ dom 𝐹)) ∪ (𝐺 ∘ ◡(inr ↾ dom 𝐺))) | |
2 | 1 | dmeqi 4812 | . 2 ⊢ dom (𝐹 ⊔d 𝐺) = dom ((𝐹 ∘ ◡(inl ↾ dom 𝐹)) ∪ (𝐺 ∘ ◡(inr ↾ dom 𝐺))) |
3 | dmun 4818 | . 2 ⊢ dom ((𝐹 ∘ ◡(inl ↾ dom 𝐹)) ∪ (𝐺 ∘ ◡(inr ↾ dom 𝐺))) = (dom (𝐹 ∘ ◡(inl ↾ dom 𝐹)) ∪ dom (𝐺 ∘ ◡(inr ↾ dom 𝐺))) | |
4 | dmco 5119 | . . . . 5 ⊢ dom (𝐹 ∘ ◡(inl ↾ dom 𝐹)) = (◡◡(inl ↾ dom 𝐹) “ dom 𝐹) | |
5 | imacnvcnv 5075 | . . . . 5 ⊢ (◡◡(inl ↾ dom 𝐹) “ dom 𝐹) = ((inl ↾ dom 𝐹) “ dom 𝐹) | |
6 | resima 4924 | . . . . . 6 ⊢ ((inl ↾ dom 𝐹) “ dom 𝐹) = (inl “ dom 𝐹) | |
7 | df-ima 4624 | . . . . . 6 ⊢ (inl “ dom 𝐹) = ran (inl ↾ dom 𝐹) | |
8 | 6, 7 | eqtri 2191 | . . . . 5 ⊢ ((inl ↾ dom 𝐹) “ dom 𝐹) = ran (inl ↾ dom 𝐹) |
9 | 4, 5, 8 | 3eqtri 2195 | . . . 4 ⊢ dom (𝐹 ∘ ◡(inl ↾ dom 𝐹)) = ran (inl ↾ dom 𝐹) |
10 | dmco 5119 | . . . . 5 ⊢ dom (𝐺 ∘ ◡(inr ↾ dom 𝐺)) = (◡◡(inr ↾ dom 𝐺) “ dom 𝐺) | |
11 | imacnvcnv 5075 | . . . . 5 ⊢ (◡◡(inr ↾ dom 𝐺) “ dom 𝐺) = ((inr ↾ dom 𝐺) “ dom 𝐺) | |
12 | resima 4924 | . . . . . 6 ⊢ ((inr ↾ dom 𝐺) “ dom 𝐺) = (inr “ dom 𝐺) | |
13 | df-ima 4624 | . . . . . 6 ⊢ (inr “ dom 𝐺) = ran (inr ↾ dom 𝐺) | |
14 | 12, 13 | eqtri 2191 | . . . . 5 ⊢ ((inr ↾ dom 𝐺) “ dom 𝐺) = ran (inr ↾ dom 𝐺) |
15 | 10, 11, 14 | 3eqtri 2195 | . . . 4 ⊢ dom (𝐺 ∘ ◡(inr ↾ dom 𝐺)) = ran (inr ↾ dom 𝐺) |
16 | 9, 15 | uneq12i 3279 | . . 3 ⊢ (dom (𝐹 ∘ ◡(inl ↾ dom 𝐹)) ∪ dom (𝐺 ∘ ◡(inr ↾ dom 𝐺))) = (ran (inl ↾ dom 𝐹) ∪ ran (inr ↾ dom 𝐺)) |
17 | djuunr 7043 | . . 3 ⊢ (ran (inl ↾ dom 𝐹) ∪ ran (inr ↾ dom 𝐺)) = (dom 𝐹 ⊔ dom 𝐺) | |
18 | 16, 17 | eqtri 2191 | . 2 ⊢ (dom (𝐹 ∘ ◡(inl ↾ dom 𝐹)) ∪ dom (𝐺 ∘ ◡(inr ↾ dom 𝐺))) = (dom 𝐹 ⊔ dom 𝐺) |
19 | 2, 3, 18 | 3eqtri 2195 | 1 ⊢ dom (𝐹 ⊔d 𝐺) = (dom 𝐹 ⊔ dom 𝐺) |
Colors of variables: wff set class |
Syntax hints: = wceq 1348 ∪ cun 3119 ◡ccnv 4610 dom cdm 4611 ran crn 4612 ↾ cres 4613 “ cima 4614 ∘ ccom 4615 ⊔ cdju 7014 inlcinl 7022 inrcinr 7023 ⊔d cdjud 7079 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 ax-un 4418 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-sbc 2956 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-br 3990 df-opab 4051 df-mpt 4052 df-tr 4088 df-id 4278 df-iord 4351 df-on 4353 df-suc 4356 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-fv 5206 df-1st 6119 df-2nd 6120 df-1o 6395 df-dju 7015 df-inl 7024 df-inr 7025 df-djud 7080 |
This theorem is referenced by: (None) |
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