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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 5245 | . . . . 5 | |
3 | 1, 2 | syl 14 | . . . 4 |
4 | casef.g | . . . . 5 | |
5 | ffun 5245 | . . . . 5 | |
6 | 4, 5 | syl 14 | . . . 4 |
7 | 3, 6 | casefun 6938 | . . 3 case |
8 | caserel 6940 | . . . 4 case ⊔ | |
9 | ssid 3087 | . . . . 5 ⊔ ⊔ | |
10 | frn 5251 | . . . . . . 7 | |
11 | 1, 10 | syl 14 | . . . . . 6 |
12 | frn 5251 | . . . . . . 7 | |
13 | 4, 12 | syl 14 | . . . . . 6 |
14 | 11, 13 | unssd 3222 | . . . . 5 |
15 | xpss12 4616 | . . . . 5 ⊔ ⊔ ⊔ ⊔ | |
16 | 9, 14, 15 | sylancr 410 | . . . 4 ⊔ ⊔ |
17 | 8, 16 | sstrid 3078 | . . 3 case ⊔ |
18 | funssxp 5262 | . . . 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 6939 | . . . 4 case ⊔ | |
22 | fdm 5248 | . . . . . 6 | |
23 | 1, 22 | syl 14 | . . . . 5 |
24 | fdm 5248 | . . . . . 6 | |
25 | 4, 24 | syl 14 | . . . . 5 |
26 | djueq12 6892 | . . . . 5 ⊔ ⊔ | |
27 | 23, 25, 26 | syl2anc 408 | . . . 4 ⊔ ⊔ |
28 | 21, 27 | syl5eq 2162 | . . 3 case ⊔ |
29 | 28 | feq2d 5230 | . 2 case case case ⊔ |
30 | 20, 29 | mpbid 146 | 1 case ⊔ |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1316 cun 3039 wss 3041 cxp 4507 cdm 4509 crn 4510 wfun 5087 wf 5089 ⊔ cdju 6890 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-fal 1322 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-ne 2286 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-ima 4522 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: casef1 6943 omp1eomlem 6947 ctm 6962 |
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