Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 7033 | . . . 4 ⊢ (inl ↾ 𝐴):𝐴–1-1-onto→({∅} × 𝐴) | |
2 | f1ofo 5449 | . . . 4 ⊢ ((inl ↾ 𝐴):𝐴–1-1-onto→({∅} × 𝐴) → (inl ↾ 𝐴):𝐴–onto→({∅} × 𝐴)) | |
3 | forn 5423 | . . . 4 ⊢ ((inl ↾ 𝐴):𝐴–onto→({∅} × 𝐴) → ran (inl ↾ 𝐴) = ({∅} × 𝐴)) | |
4 | 1, 2, 3 | mp2b 8 | . . 3 ⊢ ran (inl ↾ 𝐴) = ({∅} × 𝐴) |
5 | djurf1or 7034 | . . . 4 ⊢ (inr ↾ 𝐵):𝐵–1-1-onto→({1o} × 𝐵) | |
6 | f1ofo 5449 | . . . 4 ⊢ ((inr ↾ 𝐵):𝐵–1-1-onto→({1o} × 𝐵) → (inr ↾ 𝐵):𝐵–onto→({1o} × 𝐵)) | |
7 | forn 5423 | . . . 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 3279 | . 2 ⊢ (ran (inl ↾ 𝐴) ∪ ran (inr ↾ 𝐵)) = (({∅} × 𝐴) ∪ ({1o} × 𝐵)) |
10 | df-dju 7015 | . 2 ⊢ (𝐴 ⊔ 𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵)) | |
11 | 9, 10 | eqtr4i 2194 | 1 ⊢ (ran (inl ↾ 𝐴) ∪ ran (inr ↾ 𝐵)) = (𝐴 ⊔ 𝐵) |
Colors of variables: wff set class |
Syntax hints: = wceq 1348 ∪ cun 3119 ∅c0 3414 {csn 3583 × cxp 4609 ran crn 4612 ↾ cres 4613 –onto→wfo 5196 –1-1-onto→wf1o 5197 1oc1o 6388 ⊔ cdju 7014 inlcinl 7022 inrcinr 7023 |
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-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 |
This theorem is referenced by: djuun 7044 eldju 7045 casedm 7063 djudm 7082 |
Copyright terms: Public domain | W3C validator |