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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 6946 | . . . . . . 7 inl ⊔ | |
2 | f1fun 5331 | . . . . . . 7 inl ⊔ inl | |
3 | 1, 2 | ax-mp 5 | . . . . . 6 inl |
4 | funcnvcnv 5182 | . . . . . 6 inl inl | |
5 | 3, 4 | ax-mp 5 | . . . . 5 inl |
6 | djuinj.r | . . . . 5 | |
7 | funco 5163 | . . . . 5 inl inl | |
8 | 5, 6, 7 | sylancr 410 | . . . 4 inl |
9 | cnvco 4724 | . . . . 5 inl inl | |
10 | 9 | funeqi 5144 | . . . 4 inl inl |
11 | 8, 10 | sylibr 133 | . . 3 inl |
12 | inrresf1 6947 | . . . . . . 7 inr ⊔ | |
13 | f1fun 5331 | . . . . . . 7 inr ⊔ inr | |
14 | 12, 13 | ax-mp 5 | . . . . . 6 inr |
15 | funcnvcnv 5182 | . . . . . 6 inr inr | |
16 | 14, 15 | ax-mp 5 | . . . . 5 inr |
17 | djuinj.s | . . . . 5 | |
18 | funco 5163 | . . . . 5 inr inr | |
19 | 16, 17, 18 | sylancr 410 | . . . 4 inr |
20 | cnvco 4724 | . . . . 5 inr inr | |
21 | 20 | funeqi 5144 | . . . 4 inr inr |
22 | 19, 21 | sylibr 133 | . . 3 inr |
23 | df-rn 4550 | . . . . . . 7 inl inl | |
24 | rncoss 4809 | . . . . . . 7 inl | |
25 | 23, 24 | eqsstrri 3130 | . . . . . 6 inl |
26 | df-rn 4550 | . . . . . . 7 inr inr | |
27 | rncoss 4809 | . . . . . . 7 inr | |
28 | 26, 27 | eqsstrri 3130 | . . . . . 6 inr |
29 | ss2in 3304 | . . . . . 6 inl inr inl inr | |
30 | 25, 28, 29 | mp2an 422 | . . . . 5 inl inr |
31 | djuinj.disj | . . . . 5 | |
32 | 30, 31 | sseqtrid 3147 | . . . 4 inl inr |
33 | ss0 3403 | . . . 4 inl inr inl inr | |
34 | 32, 33 | syl 14 | . . 3 inl inr |
35 | funun 5167 | . . 3 inl inr inl inr inl inr | |
36 | 11, 22, 34, 35 | syl21anc 1215 | . 2 inl inr |
37 | df-djud 6988 | . . . . 5 ⊔d inl inr | |
38 | 37 | cnveqi 4714 | . . . 4 ⊔d inl inr |
39 | cnvun 4944 | . . . 4 inl inr inl inr | |
40 | 38, 39 | eqtri 2160 | . . 3 ⊔d inl inr |
41 | 40 | funeqi 5144 | . 2 ⊔d inl inr |
42 | 36, 41 | sylibr 133 | 1 ⊔d |
Colors of variables: wff set class |
Syntax hints: wi 4 wceq 1331 cun 3069 cin 3070 wss 3071 c0 3363 ccnv 4538 cdm 4539 crn 4540 cres 4541 ccom 4543 wfun 5117 wf1 5120 ⊔ cdju 6922 inlcinl 6930 inrcinr 6931 ⊔d cdjud 6987 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-iord 4288 df-on 4290 df-suc 4293 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-1st 6038 df-2nd 6039 df-1o 6313 df-dju 6923 df-inl 6932 df-inr 6933 df-djud 6988 |
This theorem is referenced by: (None) |
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