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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 6969 | . 2 case inl inr | |
2 | cocnvss 5064 | . . . 4 inl inl inl | |
3 | inlresf1 6946 | . . . . . 6 inl ⊔ | |
4 | f1rn 5329 | . . . . . 6 inl ⊔ inl ⊔ | |
5 | 3, 4 | ax-mp 5 | . . . . 5 inl ⊔ |
6 | resss 4843 | . . . . . . 7 inl | |
7 | rnss 4769 | . . . . . . 7 inl inl | |
8 | 6, 7 | ax-mp 5 | . . . . . 6 inl |
9 | ssun1 3239 | . . . . . 6 | |
10 | 8, 9 | sstri 3106 | . . . . 5 inl |
11 | xpss12 4646 | . . . . 5 inl ⊔ inl inl inl ⊔ | |
12 | 5, 10, 11 | mp2an 422 | . . . 4 inl inl ⊔ |
13 | 2, 12 | sstri 3106 | . . 3 inl ⊔ |
14 | cocnvss 5064 | . . . 4 inr inr inr | |
15 | inrresf1 6947 | . . . . . 6 inr ⊔ | |
16 | f1rn 5329 | . . . . . 6 inr ⊔ inr ⊔ | |
17 | 15, 16 | ax-mp 5 | . . . . 5 inr ⊔ |
18 | resss 4843 | . . . . . . 7 inr | |
19 | rnss 4769 | . . . . . . 7 inr inr | |
20 | 18, 19 | ax-mp 5 | . . . . . 6 inr |
21 | ssun2 3240 | . . . . . 6 | |
22 | 20, 21 | sstri 3106 | . . . . 5 inr |
23 | xpss12 4646 | . . . . 5 inr ⊔ inr inr inr ⊔ | |
24 | 17, 22, 23 | mp2an 422 | . . . 4 inr inr ⊔ |
25 | 14, 24 | sstri 3106 | . . 3 inr ⊔ |
26 | 13, 25 | unssi 3251 | . 2 inl inr ⊔ |
27 | 1, 26 | eqsstri 3129 | 1 case ⊔ |
Colors of variables: wff set class |
Syntax hints: cun 3069 wss 3071 cxp 4537 ccnv 4538 cdm 4539 crn 4540 cres 4541 ccom 4543 wf1 5120 ⊔ cdju 6922 inlcinl 6930 inrcinr 6931 casecdjucase 6968 |
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 df-case 6969 |
This theorem is referenced by: casef 6973 |
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