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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 6994 | . . . . . . 7 inl | |
2 | 1 | funmpt2 5212 | . . . . . 6 inl |
3 | funcnvcnv 5232 | . . . . . 6 inl inl | |
4 | 2, 3 | ax-mp 5 | . . . . 5 inl |
5 | caseinj.r | . . . . 5 | |
6 | funco 5213 | . . . . 5 inl inl | |
7 | 4, 5, 6 | sylancr 411 | . . . 4 inl |
8 | cnvco 4774 | . . . . 5 inl inl | |
9 | 8 | funeqi 5194 | . . . 4 inl inl |
10 | 7, 9 | sylibr 133 | . . 3 inl |
11 | df-inr 6995 | . . . . . . 7 inr | |
12 | 11 | funmpt2 5212 | . . . . . 6 inr |
13 | funcnvcnv 5232 | . . . . . 6 inr inr | |
14 | 12, 13 | ax-mp 5 | . . . . 5 inr |
15 | caseinj.s | . . . . 5 | |
16 | funco 5213 | . . . . 5 inr inr | |
17 | 14, 15, 16 | sylancr 411 | . . . 4 inr |
18 | cnvco 4774 | . . . . 5 inr inr | |
19 | 18 | funeqi 5194 | . . . 4 inr inr |
20 | 17, 19 | sylibr 133 | . . 3 inr |
21 | df-rn 4600 | . . . . . . 7 inl inl | |
22 | rncoss 4859 | . . . . . . 7 inl | |
23 | 21, 22 | eqsstrri 3161 | . . . . . 6 inl |
24 | df-rn 4600 | . . . . . . 7 inr inr | |
25 | rncoss 4859 | . . . . . . 7 inr | |
26 | 24, 25 | eqsstrri 3161 | . . . . . 6 inr |
27 | ss2in 3336 | . . . . . 6 inl inr inl inr | |
28 | 23, 26, 27 | mp2an 423 | . . . . 5 inl inr |
29 | caseinj.disj | . . . . 5 | |
30 | 28, 29 | sseqtrid 3178 | . . . 4 inl inr |
31 | ss0 3435 | . . . 4 inl inr inl inr | |
32 | 30, 31 | syl 14 | . . 3 inl inr |
33 | funun 5217 | . . 3 inl inr inl inr inl inr | |
34 | 10, 20, 32, 33 | syl21anc 1219 | . 2 inl inr |
35 | df-case 7031 | . . . . 5 case inl inr | |
36 | 35 | cnveqi 4764 | . . . 4 case inl inr |
37 | cnvun 4994 | . . . 4 inl inr inl inr | |
38 | 36, 37 | eqtri 2178 | . . 3 case inl inr |
39 | 38 | funeqi 5194 | . 2 case inl inr |
40 | 34, 39 | sylibr 133 | 1 case |
Colors of variables: wff set class |
Syntax hints: wi 4 wceq 1335 cvv 2712 cun 3100 cin 3101 wss 3102 c0 3395 cop 3564 ccnv 4588 cdm 4589 crn 4590 ccom 4593 wfun 5167 c1o 6359 inlcinl 6992 inrcinr 6993 casecdjucase 7030 |
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 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-14 2131 ax-ext 2139 ax-sep 4085 ax-pow 4138 ax-pr 4172 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-v 2714 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3396 df-pw 3546 df-sn 3567 df-pr 3568 df-op 3570 df-br 3968 df-opab 4029 df-mpt 4030 df-id 4256 df-xp 4595 df-rel 4596 df-cnv 4597 df-co 4598 df-dm 4599 df-rn 4600 df-fun 5175 df-inl 6994 df-inr 6995 df-case 7031 |
This theorem is referenced by: casef1 7037 |
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