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Mirrors > Home > ILE Home > Th. List > caserel | Unicode version |
Description: The "case" construction of two relations is a relation, with bounds on its domain and codomain. Typically, the "case" construction is used when both relations have a common codomain. (Contributed by BJ, 10-Jul-2022.) |
Ref | Expression |
---|---|
caserel | case ⊔ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-case 6937 | . 2 case inl inr | |
2 | cocnvss 5034 | . . . 4 inl inl inl | |
3 | inlresf1 6914 | . . . . . 6 inl ⊔ | |
4 | f1rn 5299 | . . . . . 6 inl ⊔ inl ⊔ | |
5 | 3, 4 | ax-mp 5 | . . . . 5 inl ⊔ |
6 | resss 4813 | . . . . . . 7 inl | |
7 | rnss 4739 | . . . . . . 7 inl inl | |
8 | 6, 7 | ax-mp 5 | . . . . . 6 inl |
9 | ssun1 3209 | . . . . . 6 | |
10 | 8, 9 | sstri 3076 | . . . . 5 inl |
11 | xpss12 4616 | . . . . 5 inl ⊔ inl inl inl ⊔ | |
12 | 5, 10, 11 | mp2an 422 | . . . 4 inl inl ⊔ |
13 | 2, 12 | sstri 3076 | . . 3 inl ⊔ |
14 | cocnvss 5034 | . . . 4 inr inr inr | |
15 | inrresf1 6915 | . . . . . 6 inr ⊔ | |
16 | f1rn 5299 | . . . . . 6 inr ⊔ inr ⊔ | |
17 | 15, 16 | ax-mp 5 | . . . . 5 inr ⊔ |
18 | resss 4813 | . . . . . . 7 inr | |
19 | rnss 4739 | . . . . . . 7 inr inr | |
20 | 18, 19 | ax-mp 5 | . . . . . 6 inr |
21 | ssun2 3210 | . . . . . 6 | |
22 | 20, 21 | sstri 3076 | . . . . 5 inr |
23 | xpss12 4616 | . . . . 5 inr ⊔ inr inr inr ⊔ | |
24 | 17, 22, 23 | mp2an 422 | . . . 4 inr inr ⊔ |
25 | 14, 24 | sstri 3076 | . . 3 inr ⊔ |
26 | 13, 25 | unssi 3221 | . 2 inl inr ⊔ |
27 | 1, 26 | eqsstri 3099 | 1 case ⊔ |
Colors of variables: wff set class |
Syntax hints: cun 3039 wss 3041 cxp 4507 ccnv 4508 cdm 4509 crn 4510 cres 4511 ccom 4513 wf1 5090 ⊔ cdju 6890 inlcinl 6898 inrcinr 6899 casecdjucase 6936 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-nul 4024 ax-pow 4068 ax-pr 4101 ax-un 4325 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-sbc 2883 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-mpt 3961 df-tr 3997 df-id 4185 df-iord 4258 df-on 4260 df-suc 4263 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-f1 5098 df-fo 5099 df-f1o 5100 df-fv 5101 df-1st 6006 df-2nd 6007 df-1o 6281 df-dju 6891 df-inl 6900 df-inr 6901 df-case 6937 |
This theorem is referenced by: casef 6941 |
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