Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > casef1 | Unicode version |
Description: The "case" construction of two injective functions with disjoint ranges is an injective function. (Contributed by BJ, 10-Jul-2022.) |
Ref | Expression |
---|---|
casef1.f | |
casef1.g | |
casef1.disj |
Ref | Expression |
---|---|
casef1 | case ⊔ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | casef1.f | . . . 4 | |
2 | f1f 5298 | . . . 4 | |
3 | 1, 2 | syl 14 | . . 3 |
4 | casef1.g | . . . 4 | |
5 | f1f 5298 | . . . 4 | |
6 | 4, 5 | syl 14 | . . 3 |
7 | 3, 6 | casef 6941 | . 2 case ⊔ |
8 | df-f1 5098 | . . . . 5 | |
9 | 8 | simprbi 273 | . . . 4 |
10 | 1, 9 | syl 14 | . . 3 |
11 | df-f1 5098 | . . . . 5 | |
12 | 11 | simprbi 273 | . . . 4 |
13 | 4, 12 | syl 14 | . . 3 |
14 | casef1.disj | . . 3 | |
15 | 10, 13, 14 | caseinj 6942 | . 2 case |
16 | df-f1 5098 | . 2 case ⊔ case ⊔ case | |
17 | 7, 15, 16 | sylanbrc 413 | 1 case ⊔ |
Colors of variables: wff set class |
Syntax hints: wi 4 wceq 1316 cin 3040 c0 3333 ccnv 4508 crn 4510 wfun 5087 wf 5089 wf1 5090 ⊔ 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: djudom 6946 exmidsbthrlem 13113 |
Copyright terms: Public domain | W3C validator |