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Mirrors > Home > ILE Home > Th. List > f1dmvrnfibi | Unicode version |
Description: A one-to-one function whose domain is a set is finite if and only if its range is finite. See also f1vrnfibi 6801. (Contributed by AV, 10-Jan-2020.) |
Ref | Expression |
---|---|
f1dmvrnfibi |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1rel 5302 | . . . 4 | |
2 | 1 | ad2antlr 480 | . . 3 |
3 | f1cnv 5359 | . . . . 5 | |
4 | f1ofun 5337 | . . . . 5 | |
5 | 3, 4 | syl 14 | . . . 4 |
6 | 5 | ad2antlr 480 | . . 3 |
7 | simpr 109 | . . 3 | |
8 | funrnfi 6798 | . . 3 | |
9 | 2, 6, 7, 8 | syl3anc 1201 | . 2 |
10 | simpr 109 | . . . 4 | |
11 | f1dm 5303 | . . . . . . . 8 | |
12 | f1f1orn 5346 | . . . . . . . 8 | |
13 | eleq1 2180 | . . . . . . . . . . . 12 | |
14 | f1oeq2 5327 | . . . . . . . . . . . 12 | |
15 | 13, 14 | anbi12d 464 | . . . . . . . . . . 11 |
16 | 15 | eqcoms 2120 | . . . . . . . . . 10 |
17 | 16 | biimpd 143 | . . . . . . . . 9 |
18 | 17 | expcomd 1402 | . . . . . . . 8 |
19 | 11, 12, 18 | sylc 62 | . . . . . . 7 |
20 | 19 | impcom 124 | . . . . . 6 |
21 | 20 | adantr 274 | . . . . 5 |
22 | f1oeng 6619 | . . . . 5 | |
23 | 21, 22 | syl 14 | . . . 4 |
24 | enfii 6736 | . . . 4 | |
25 | 10, 23, 24 | syl2anc 408 | . . 3 |
26 | f1fun 5301 | . . . . 5 | |
27 | 26 | ad2antlr 480 | . . . 4 |
28 | fundmfibi 6795 | . . . 4 | |
29 | 27, 28 | syl 14 | . . 3 |
30 | 25, 29 | mpbird 166 | . 2 |
31 | 9, 30 | impbida 570 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1316 wcel 1465 class class class wbr 3899 ccnv 4508 cdm 4509 crn 4510 wrel 4514 wfun 5087 wf1 5090 wf1o 5092 cen 6600 cfn 6602 |
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-coll 4013 ax-sep 4016 ax-nul 4024 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-iinf 4472 |
This theorem depends on definitions: df-bi 116 df-dc 805 df-3or 948 df-3an 949 df-tru 1319 df-fal 1322 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-ne 2286 df-ral 2398 df-rex 2399 df-reu 2400 df-rab 2402 df-v 2662 df-sbc 2883 df-csb 2976 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-if 3445 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-int 3742 df-iun 3785 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-iom 4475 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 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-er 6397 df-en 6603 df-fin 6605 |
This theorem is referenced by: f1vrnfibi 6801 |
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