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Mirrors > Home > ILE Home > Th. List > djuin | GIF version |
Description: The images of any classes under right and left injection produce disjoint sets. (Contributed by Jim Kingdon, 21-Jun-2022.) (Proof shortened by BJ, 9-Jul-2023.) |
Ref | Expression |
---|---|
djuin | ⊢ ((inl “ 𝐴) ∩ (inr “ 𝐵)) = ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ima 4611 | . . 3 ⊢ (inl “ 𝐴) = ran (inl ↾ 𝐴) | |
2 | df-ima 4611 | . . 3 ⊢ (inr “ 𝐵) = ran (inr ↾ 𝐵) | |
3 | 1, 2 | ineq12i 3316 | . 2 ⊢ ((inl “ 𝐴) ∩ (inr “ 𝐵)) = (ran (inl ↾ 𝐴) ∩ ran (inr ↾ 𝐵)) |
4 | djuinr 7019 | . 2 ⊢ (ran (inl ↾ 𝐴) ∩ ran (inr ↾ 𝐵)) = ∅ | |
5 | 3, 4 | eqtri 2185 | 1 ⊢ ((inl “ 𝐴) ∩ (inr “ 𝐵)) = ∅ |
Colors of variables: wff set class |
Syntax hints: = wceq 1342 ∩ cin 3110 ∅c0 3404 ran crn 4599 ↾ cres 4600 “ cima 4601 inlcinl 7001 inrcinr 7002 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-nul 4102 ax-pow 4147 ax-pr 4181 ax-un 4405 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-ral 2447 df-rex 2448 df-v 2723 df-sbc 2947 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-opab 4038 df-mpt 4039 df-tr 4075 df-id 4265 df-iord 4338 df-on 4340 df-suc 4343 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-1st 6100 df-2nd 6101 df-1o 6375 df-inl 7003 df-inr 7004 |
This theorem is referenced by: caseinl 7047 caseinr 7048 endjusym 7052 ctssdccl 7067 dju1p1e2 7144 endjudisj 7157 djuen 7158 |
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