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Mirrors > Home > ILE Home > Th. List > djuunr | GIF version |
Description: The disjoint union of two classes is the union of the images of those two classes under right and left injection. (Contributed by Jim Kingdon, 22-Jun-2022.) (Proof shortened by BJ, 6-Jul-2022.) |
Ref | Expression |
---|---|
djuunr | ⊢ (ran (inl ↾ 𝐴) ∪ ran (inr ↾ 𝐵)) = (𝐴 ⊔ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | djulf1or 7000 | . . . 4 ⊢ (inl ↾ 𝐴):𝐴–1-1-onto→({∅} × 𝐴) | |
2 | f1ofo 5421 | . . . 4 ⊢ ((inl ↾ 𝐴):𝐴–1-1-onto→({∅} × 𝐴) → (inl ↾ 𝐴):𝐴–onto→({∅} × 𝐴)) | |
3 | forn 5395 | . . . 4 ⊢ ((inl ↾ 𝐴):𝐴–onto→({∅} × 𝐴) → ran (inl ↾ 𝐴) = ({∅} × 𝐴)) | |
4 | 1, 2, 3 | mp2b 8 | . . 3 ⊢ ran (inl ↾ 𝐴) = ({∅} × 𝐴) |
5 | djurf1or 7001 | . . . 4 ⊢ (inr ↾ 𝐵):𝐵–1-1-onto→({1o} × 𝐵) | |
6 | f1ofo 5421 | . . . 4 ⊢ ((inr ↾ 𝐵):𝐵–1-1-onto→({1o} × 𝐵) → (inr ↾ 𝐵):𝐵–onto→({1o} × 𝐵)) | |
7 | forn 5395 | . . . 4 ⊢ ((inr ↾ 𝐵):𝐵–onto→({1o} × 𝐵) → ran (inr ↾ 𝐵) = ({1o} × 𝐵)) | |
8 | 5, 6, 7 | mp2b 8 | . . 3 ⊢ ran (inr ↾ 𝐵) = ({1o} × 𝐵) |
9 | 4, 8 | uneq12i 3259 | . 2 ⊢ (ran (inl ↾ 𝐴) ∪ ran (inr ↾ 𝐵)) = (({∅} × 𝐴) ∪ ({1o} × 𝐵)) |
10 | df-dju 6982 | . 2 ⊢ (𝐴 ⊔ 𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵)) | |
11 | 9, 10 | eqtr4i 2181 | 1 ⊢ (ran (inl ↾ 𝐴) ∪ ran (inr ↾ 𝐵)) = (𝐴 ⊔ 𝐵) |
Colors of variables: wff set class |
Syntax hints: = wceq 1335 ∪ cun 3100 ∅c0 3394 {csn 3560 × cxp 4584 ran crn 4587 ↾ cres 4588 –onto→wfo 5168 –1-1-onto→wf1o 5169 1oc1o 6356 ⊔ cdju 6981 inlcinl 6989 inrcinr 6990 |
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 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4082 ax-nul 4090 ax-pow 4135 ax-pr 4169 ax-un 4393 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-v 2714 df-sbc 2938 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3395 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-br 3966 df-opab 4026 df-mpt 4027 df-tr 4063 df-id 4253 df-iord 4326 df-on 4328 df-suc 4331 df-xp 4592 df-rel 4593 df-cnv 4594 df-co 4595 df-dm 4596 df-rn 4597 df-res 4598 df-iota 5135 df-fun 5172 df-fn 5173 df-f 5174 df-f1 5175 df-fo 5176 df-f1o 5177 df-fv 5178 df-1st 6088 df-2nd 6089 df-1o 6363 df-dju 6982 df-inl 6991 df-inr 6992 |
This theorem is referenced by: djuun 7011 eldju 7012 casedm 7030 djudm 7049 |
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