Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > djur | GIF version |
Description: A member of a disjoint union can be mapped from one of the classes which produced it. (Contributed by Jim Kingdon, 23-Jun-2022.) Upgrade implication to biconditional and shorten proof. (Revised by BJ, 14-Jul-2023.) |
Ref | Expression |
---|---|
djur | ⊢ (𝐶 ∈ (𝐴 ⊔ 𝐵) ↔ (∃𝑥 ∈ 𝐴 𝐶 = (inl‘𝑥) ∨ ∃𝑥 ∈ 𝐵 𝐶 = (inr‘𝑥))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eldju 6953 | . 2 ⊢ (𝐶 ∈ (𝐴 ⊔ 𝐵) ↔ (∃𝑥 ∈ 𝐴 𝐶 = ((inl ↾ 𝐴)‘𝑥) ∨ ∃𝑥 ∈ 𝐵 𝐶 = ((inr ↾ 𝐵)‘𝑥))) | |
2 | fvres 5445 | . . . . 5 ⊢ (𝑥 ∈ 𝐴 → ((inl ↾ 𝐴)‘𝑥) = (inl‘𝑥)) | |
3 | 2 | eqeq2d 2151 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → (𝐶 = ((inl ↾ 𝐴)‘𝑥) ↔ 𝐶 = (inl‘𝑥))) |
4 | 3 | rexbiia 2450 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 𝐶 = ((inl ↾ 𝐴)‘𝑥) ↔ ∃𝑥 ∈ 𝐴 𝐶 = (inl‘𝑥)) |
5 | fvres 5445 | . . . . 5 ⊢ (𝑥 ∈ 𝐵 → ((inr ↾ 𝐵)‘𝑥) = (inr‘𝑥)) | |
6 | 5 | eqeq2d 2151 | . . . 4 ⊢ (𝑥 ∈ 𝐵 → (𝐶 = ((inr ↾ 𝐵)‘𝑥) ↔ 𝐶 = (inr‘𝑥))) |
7 | 6 | rexbiia 2450 | . . 3 ⊢ (∃𝑥 ∈ 𝐵 𝐶 = ((inr ↾ 𝐵)‘𝑥) ↔ ∃𝑥 ∈ 𝐵 𝐶 = (inr‘𝑥)) |
8 | 4, 7 | orbi12i 753 | . 2 ⊢ ((∃𝑥 ∈ 𝐴 𝐶 = ((inl ↾ 𝐴)‘𝑥) ∨ ∃𝑥 ∈ 𝐵 𝐶 = ((inr ↾ 𝐵)‘𝑥)) ↔ (∃𝑥 ∈ 𝐴 𝐶 = (inl‘𝑥) ∨ ∃𝑥 ∈ 𝐵 𝐶 = (inr‘𝑥))) |
9 | 1, 8 | bitri 183 | 1 ⊢ (𝐶 ∈ (𝐴 ⊔ 𝐵) ↔ (∃𝑥 ∈ 𝐴 𝐶 = (inl‘𝑥) ∨ ∃𝑥 ∈ 𝐵 𝐶 = (inr‘𝑥))) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 ∨ wo 697 = wceq 1331 ∈ wcel 1480 ∃wrex 2417 ↾ cres 4541 ‘cfv 5123 ⊔ cdju 6922 inlcinl 6930 inrcinr 6931 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-iord 4288 df-on 4290 df-suc 4293 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-1st 6038 df-2nd 6039 df-1o 6313 df-dju 6923 df-inl 6932 df-inr 6933 |
This theorem is referenced by: djuss 6955 updjud 6967 omp1eomlem 6979 0ct 6992 ctmlemr 6993 ctssdclemn0 6995 fodjuomnilemdc 7016 exmidfodomrlemeldju 7055 |
Copyright terms: Public domain | W3C validator |