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Mirrors > Home > ILE Home > Th. List > djuinj | Unicode version |
Description: The "domain-disjoint-union" of two injective relations with disjoint ranges is an injective relation. (Contributed by BJ, 10-Jul-2022.) |
Ref | Expression |
---|---|
djuinj.r | |
djuinj.s | |
djuinj.disj |
Ref | Expression |
---|---|
djuinj | ⊔d |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | inlresf1 6914 | . . . . . . 7 inl ⊔ | |
2 | f1fun 5301 | . . . . . . 7 inl ⊔ inl | |
3 | 1, 2 | ax-mp 5 | . . . . . 6 inl |
4 | funcnvcnv 5152 | . . . . . 6 inl inl | |
5 | 3, 4 | ax-mp 5 | . . . . 5 inl |
6 | djuinj.r | . . . . 5 | |
7 | funco 5133 | . . . . 5 inl inl | |
8 | 5, 6, 7 | sylancr 410 | . . . 4 inl |
9 | cnvco 4694 | . . . . 5 inl inl | |
10 | 9 | funeqi 5114 | . . . 4 inl inl |
11 | 8, 10 | sylibr 133 | . . 3 inl |
12 | inrresf1 6915 | . . . . . . 7 inr ⊔ | |
13 | f1fun 5301 | . . . . . . 7 inr ⊔ inr | |
14 | 12, 13 | ax-mp 5 | . . . . . 6 inr |
15 | funcnvcnv 5152 | . . . . . 6 inr inr | |
16 | 14, 15 | ax-mp 5 | . . . . 5 inr |
17 | djuinj.s | . . . . 5 | |
18 | funco 5133 | . . . . 5 inr inr | |
19 | 16, 17, 18 | sylancr 410 | . . . 4 inr |
20 | cnvco 4694 | . . . . 5 inr inr | |
21 | 20 | funeqi 5114 | . . . 4 inr inr |
22 | 19, 21 | sylibr 133 | . . 3 inr |
23 | df-rn 4520 | . . . . . . 7 inl inl | |
24 | rncoss 4779 | . . . . . . 7 inl | |
25 | 23, 24 | eqsstrri 3100 | . . . . . 6 inl |
26 | df-rn 4520 | . . . . . . 7 inr inr | |
27 | rncoss 4779 | . . . . . . 7 inr | |
28 | 26, 27 | eqsstrri 3100 | . . . . . 6 inr |
29 | ss2in 3274 | . . . . . 6 inl inr inl inr | |
30 | 25, 28, 29 | mp2an 422 | . . . . 5 inl inr |
31 | djuinj.disj | . . . . 5 | |
32 | 30, 31 | sseqtrid 3117 | . . . 4 inl inr |
33 | ss0 3373 | . . . 4 inl inr inl inr | |
34 | 32, 33 | syl 14 | . . 3 inl inr |
35 | funun 5137 | . . 3 inl inr inl inr inl inr | |
36 | 11, 22, 34, 35 | syl21anc 1200 | . 2 inl inr |
37 | df-djud 6956 | . . . . 5 ⊔d inl inr | |
38 | 37 | cnveqi 4684 | . . . 4 ⊔d inl inr |
39 | cnvun 4914 | . . . 4 inl inr inl inr | |
40 | 38, 39 | eqtri 2138 | . . 3 ⊔d inl inr |
41 | 40 | funeqi 5114 | . 2 ⊔d inl inr |
42 | 36, 41 | sylibr 133 | 1 ⊔d |
Colors of variables: wff set class |
Syntax hints: wi 4 wceq 1316 cun 3039 cin 3040 wss 3041 c0 3333 ccnv 4508 cdm 4509 crn 4510 cres 4511 ccom 4513 wfun 5087 wf1 5090 ⊔ cdju 6890 inlcinl 6898 inrcinr 6899 ⊔d cdjud 6955 |
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-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-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-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-djud 6956 |
This theorem is referenced by: (None) |
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