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Mirrors > Home > ILE Home > Th. List > djuun | 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, 9-Jul-2023.) |
Ref | Expression |
---|---|
djuun | ⊢ ((inl “ 𝐴) ∪ (inr “ 𝐵)) = (𝐴 ⊔ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ima 4635 | . . 3 ⊢ (inl “ 𝐴) = ran (inl ↾ 𝐴) | |
2 | df-ima 4635 | . . 3 ⊢ (inr “ 𝐵) = ran (inr ↾ 𝐵) | |
3 | 1, 2 | uneq12i 3287 | . 2 ⊢ ((inl “ 𝐴) ∪ (inr “ 𝐵)) = (ran (inl ↾ 𝐴) ∪ ran (inr ↾ 𝐵)) |
4 | djuunr 7058 | . 2 ⊢ (ran (inl ↾ 𝐴) ∪ ran (inr ↾ 𝐵)) = (𝐴 ⊔ 𝐵) | |
5 | 3, 4 | eqtri 2198 | 1 ⊢ ((inl “ 𝐴) ∪ (inr “ 𝐵)) = (𝐴 ⊔ 𝐵) |
Colors of variables: wff set class |
Syntax hints: = wceq 1353 ∪ cun 3127 ran crn 4623 ↾ cres 4624 “ cima 4625 ⊔ cdju 7029 inlcinl 7037 inrcinr 7038 |
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 4118 ax-nul 4126 ax-pow 4171 ax-pr 4205 ax-un 4429 |
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 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-br 4001 df-opab 4062 df-mpt 4063 df-tr 4099 df-id 4289 df-iord 4362 df-on 4364 df-suc 4367 df-xp 4628 df-rel 4629 df-cnv 4630 df-co 4631 df-dm 4632 df-rn 4633 df-res 4634 df-ima 4635 df-iota 5173 df-fun 5213 df-fn 5214 df-f 5215 df-f1 5216 df-fo 5217 df-f1o 5218 df-fv 5219 df-1st 6134 df-2nd 6135 df-1o 6410 df-dju 7030 df-inl 7039 df-inr 7040 |
This theorem is referenced by: endjusym 7088 ctssdccl 7103 dju1p1e2 7189 endjudisj 7202 djuen 7203 |
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