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Mirrors > Home > ILE Home > Th. List > caseinj | Unicode version |
Description: The "case" construction of two injective relations with disjoint ranges is an injective relation. (Contributed by BJ, 10-Jul-2022.) |
Ref | Expression |
---|---|
caseinj.r | |
caseinj.s | |
caseinj.disj |
Ref | Expression |
---|---|
caseinj | case |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-inl 6900 | . . . . . . 7 inl | |
2 | 1 | funmpt2 5132 | . . . . . 6 inl |
3 | funcnvcnv 5152 | . . . . . 6 inl inl | |
4 | 2, 3 | ax-mp 5 | . . . . 5 inl |
5 | caseinj.r | . . . . 5 | |
6 | funco 5133 | . . . . 5 inl inl | |
7 | 4, 5, 6 | sylancr 410 | . . . 4 inl |
8 | cnvco 4694 | . . . . 5 inl inl | |
9 | 8 | funeqi 5114 | . . . 4 inl inl |
10 | 7, 9 | sylibr 133 | . . 3 inl |
11 | df-inr 6901 | . . . . . . 7 inr | |
12 | 11 | funmpt2 5132 | . . . . . 6 inr |
13 | funcnvcnv 5152 | . . . . . 6 inr inr | |
14 | 12, 13 | ax-mp 5 | . . . . 5 inr |
15 | caseinj.s | . . . . 5 | |
16 | funco 5133 | . . . . 5 inr inr | |
17 | 14, 15, 16 | sylancr 410 | . . . 4 inr |
18 | cnvco 4694 | . . . . 5 inr inr | |
19 | 18 | funeqi 5114 | . . . 4 inr inr |
20 | 17, 19 | sylibr 133 | . . 3 inr |
21 | df-rn 4520 | . . . . . . 7 inl inl | |
22 | rncoss 4779 | . . . . . . 7 inl | |
23 | 21, 22 | eqsstrri 3100 | . . . . . 6 inl |
24 | df-rn 4520 | . . . . . . 7 inr inr | |
25 | rncoss 4779 | . . . . . . 7 inr | |
26 | 24, 25 | eqsstrri 3100 | . . . . . 6 inr |
27 | ss2in 3274 | . . . . . 6 inl inr inl inr | |
28 | 23, 26, 27 | mp2an 422 | . . . . 5 inl inr |
29 | caseinj.disj | . . . . 5 | |
30 | 28, 29 | sseqtrid 3117 | . . . 4 inl inr |
31 | ss0 3373 | . . . 4 inl inr inl inr | |
32 | 30, 31 | syl 14 | . . 3 inl inr |
33 | funun 5137 | . . 3 inl inr inl inr inl inr | |
34 | 10, 20, 32, 33 | syl21anc 1200 | . 2 inl inr |
35 | df-case 6937 | . . . . 5 case inl inr | |
36 | 35 | cnveqi 4684 | . . . 4 case inl inr |
37 | cnvun 4914 | . . . 4 inl inr inl inr | |
38 | 36, 37 | eqtri 2138 | . . 3 case inl inr |
39 | 38 | funeqi 5114 | . 2 case inl inr |
40 | 34, 39 | sylibr 133 | 1 case |
Colors of variables: wff set class |
Syntax hints: wi 4 wceq 1316 cvv 2660 cun 3039 cin 3040 wss 3041 c0 3333 cop 3500 ccnv 4508 cdm 4509 crn 4510 ccom 4513 wfun 5087 c1o 6274 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-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 |
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-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-br 3900 df-opab 3960 df-mpt 3961 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-fun 5095 df-inl 6900 df-inr 6901 df-case 6937 |
This theorem is referenced by: casef1 6943 |
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