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Mirrors > Home > ILE Home > Th. List > inl11 | Unicode version |
Description: Left injection is one-to-one. (Contributed by Jim Kingdon, 12-Jul-2023.) |
Ref | Expression |
---|---|
inl11 | inl inl |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-inl 7003 | . . . 4 inl | |
2 | opeq2 3753 | . . . 4 | |
3 | elex 2732 | . . . . 5 | |
4 | 3 | adantr 274 | . . . 4 |
5 | 0ex 4103 | . . . . 5 | |
6 | simpl 108 | . . . . 5 | |
7 | opexg 4200 | . . . . 5 | |
8 | 5, 6, 7 | sylancr 411 | . . . 4 |
9 | 1, 2, 4, 8 | fvmptd3 5573 | . . 3 inl |
10 | opeq2 3753 | . . . 4 | |
11 | elex 2732 | . . . . 5 | |
12 | 11 | adantl 275 | . . . 4 |
13 | 5 | a1i 9 | . . . . 5 |
14 | opexg 4200 | . . . . 5 | |
15 | 13, 14 | sylancom 417 | . . . 4 |
16 | 1, 10, 12, 15 | fvmptd3 5573 | . . 3 inl |
17 | 9, 16 | eqeq12d 2179 | . 2 inl inl |
18 | opthg 4210 | . . . . 5 | |
19 | 5, 18 | mpan 421 | . . . 4 |
20 | eqid 2164 | . . . . 5 | |
21 | 20 | biantrur 301 | . . . 4 |
22 | 19, 21 | bitr4di 197 | . . 3 |
23 | 22 | adantr 274 | . 2 |
24 | 17, 23 | bitrd 187 | 1 inl inl |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1342 wcel 2135 cvv 2721 c0 3404 cop 3573 cfv 5182 inlcinl 7001 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-nul 4102 ax-pow 4147 ax-pr 4181 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-opab 4038 df-mpt 4039 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-iota 5147 df-fun 5184 df-fv 5190 df-inl 7003 |
This theorem is referenced by: omp1eomlem 7050 difinfsnlem 7055 |
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