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Mirrors > Home > ILE Home > Th. List > casef | Unicode version |
Description: The "case" construction of two functions is a function on the disjoint union of their domains. (Contributed by BJ, 10-Jul-2022.) |
Ref | Expression |
---|---|
casef.f | |
casef.g |
Ref | Expression |
---|---|
casef | case ⊔ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | casef.f | . . . . 5 | |
2 | ffun 5275 | . . . . 5 | |
3 | 1, 2 | syl 14 | . . . 4 |
4 | casef.g | . . . . 5 | |
5 | ffun 5275 | . . . . 5 | |
6 | 4, 5 | syl 14 | . . . 4 |
7 | 3, 6 | casefun 6970 | . . 3 case |
8 | caserel 6972 | . . . 4 case ⊔ | |
9 | ssid 3117 | . . . . 5 ⊔ ⊔ | |
10 | frn 5281 | . . . . . . 7 | |
11 | 1, 10 | syl 14 | . . . . . 6 |
12 | frn 5281 | . . . . . . 7 | |
13 | 4, 12 | syl 14 | . . . . . 6 |
14 | 11, 13 | unssd 3252 | . . . . 5 |
15 | xpss12 4646 | . . . . 5 ⊔ ⊔ ⊔ ⊔ | |
16 | 9, 14, 15 | sylancr 410 | . . . 4 ⊔ ⊔ |
17 | 8, 16 | sstrid 3108 | . . 3 case ⊔ |
18 | funssxp 5292 | . . . 4 case case ⊔ case case case ⊔ | |
19 | 18 | simplbi 272 | . . 3 case case ⊔ case case |
20 | 7, 17, 19 | syl2anc 408 | . 2 case case |
21 | casedm 6971 | . . . 4 case ⊔ | |
22 | fdm 5278 | . . . . . 6 | |
23 | 1, 22 | syl 14 | . . . . 5 |
24 | fdm 5278 | . . . . . 6 | |
25 | 4, 24 | syl 14 | . . . . 5 |
26 | djueq12 6924 | . . . . 5 ⊔ ⊔ | |
27 | 23, 25, 26 | syl2anc 408 | . . . 4 ⊔ ⊔ |
28 | 21, 27 | syl5eq 2184 | . . 3 case ⊔ |
29 | 28 | feq2d 5260 | . 2 case case case ⊔ |
30 | 20, 29 | mpbid 146 | 1 case ⊔ |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1331 cun 3069 wss 3071 cxp 4537 cdm 4539 crn 4540 wfun 5117 wf 5119 ⊔ cdju 6922 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-fal 1337 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-ne 2309 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-ima 4552 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: casef1 6975 omp1eomlem 6979 ctm 6994 |
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