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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 7024 | . . . . . . 7 inl | |
2 | 1 | funmpt2 5237 | . . . . . 6 inl |
3 | funcnvcnv 5257 | . . . . . 6 inl inl | |
4 | 2, 3 | ax-mp 5 | . . . . 5 inl |
5 | caseinj.r | . . . . 5 | |
6 | funco 5238 | . . . . 5 inl inl | |
7 | 4, 5, 6 | sylancr 412 | . . . 4 inl |
8 | cnvco 4796 | . . . . 5 inl inl | |
9 | 8 | funeqi 5219 | . . . 4 inl inl |
10 | 7, 9 | sylibr 133 | . . 3 inl |
11 | df-inr 7025 | . . . . . . 7 inr | |
12 | 11 | funmpt2 5237 | . . . . . 6 inr |
13 | funcnvcnv 5257 | . . . . . 6 inr inr | |
14 | 12, 13 | ax-mp 5 | . . . . 5 inr |
15 | caseinj.s | . . . . 5 | |
16 | funco 5238 | . . . . 5 inr inr | |
17 | 14, 15, 16 | sylancr 412 | . . . 4 inr |
18 | cnvco 4796 | . . . . 5 inr inr | |
19 | 18 | funeqi 5219 | . . . 4 inr inr |
20 | 17, 19 | sylibr 133 | . . 3 inr |
21 | df-rn 4622 | . . . . . . 7 inl inl | |
22 | rncoss 4881 | . . . . . . 7 inl | |
23 | 21, 22 | eqsstrri 3180 | . . . . . 6 inl |
24 | df-rn 4622 | . . . . . . 7 inr inr | |
25 | rncoss 4881 | . . . . . . 7 inr | |
26 | 24, 25 | eqsstrri 3180 | . . . . . 6 inr |
27 | ss2in 3355 | . . . . . 6 inl inr inl inr | |
28 | 23, 26, 27 | mp2an 424 | . . . . 5 inl inr |
29 | caseinj.disj | . . . . 5 | |
30 | 28, 29 | sseqtrid 3197 | . . . 4 inl inr |
31 | ss0 3455 | . . . 4 inl inr inl inr | |
32 | 30, 31 | syl 14 | . . 3 inl inr |
33 | funun 5242 | . . 3 inl inr inl inr inl inr | |
34 | 10, 20, 32, 33 | syl21anc 1232 | . 2 inl inr |
35 | df-case 7061 | . . . . 5 case inl inr | |
36 | 35 | cnveqi 4786 | . . . 4 case inl inr |
37 | cnvun 5016 | . . . 4 inl inr inl inr | |
38 | 36, 37 | eqtri 2191 | . . 3 case inl inr |
39 | 38 | funeqi 5219 | . 2 case inl inr |
40 | 34, 39 | sylibr 133 | 1 case |
Colors of variables: wff set class |
Syntax hints: wi 4 wceq 1348 cvv 2730 cun 3119 cin 3120 wss 3121 c0 3414 cop 3586 ccnv 4610 cdm 4611 crn 4612 ccom 4615 wfun 5192 c1o 6388 inlcinl 7022 inrcinr 7023 casecdjucase 7060 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-br 3990 df-opab 4051 df-mpt 4052 df-id 4278 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-fun 5200 df-inl 7024 df-inr 7025 df-case 7061 |
This theorem is referenced by: casef1 7067 |
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