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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 6394. (Contributed by AV, 10-Jan-2020.) |
Ref | Expression |
---|---|
f1dmvrnfibi |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1rel 5115 | . . . 4 | |
2 | 1 | ad2antlr 472 | . . 3 |
3 | f1cnv 5170 | . . . . 5 | |
4 | f1ofun 5148 | . . . . 5 | |
5 | 3, 4 | syl 14 | . . . 4 |
6 | 5 | ad2antlr 472 | . . 3 |
7 | simpr 108 | . . 3 | |
8 | funrnfi 6392 | . . 3 | |
9 | 2, 6, 7, 8 | syl3anc 1169 | . 2 |
10 | simpr 108 | . . . 4 | |
11 | f1dm 5116 | . . . . . . . 8 | |
12 | f1f1orn 5157 | . . . . . . . 8 | |
13 | eleq1 2141 | . . . . . . . . . . . 12 | |
14 | f1oeq2 5138 | . . . . . . . . . . . 12 | |
15 | 13, 14 | anbi12d 456 | . . . . . . . . . . 11 |
16 | 15 | eqcoms 2084 | . . . . . . . . . 10 |
17 | 16 | biimpd 142 | . . . . . . . . 9 |
18 | 17 | expcomd 1370 | . . . . . . . 8 |
19 | 11, 12, 18 | sylc 61 | . . . . . . 7 |
20 | 19 | impcom 123 | . . . . . 6 |
21 | 20 | adantr 270 | . . . . 5 |
22 | f1oeng 6260 | . . . . 5 | |
23 | 21, 22 | syl 14 | . . . 4 |
24 | enfii 6359 | . . . 4 | |
25 | 10, 23, 24 | syl2anc 403 | . . 3 |
26 | f1fun 5114 | . . . . 5 | |
27 | 26 | ad2antlr 472 | . . . 4 |
28 | fundmfibi 6390 | . . . 4 | |
29 | 27, 28 | syl 14 | . . 3 |
30 | 25, 29 | mpbird 165 | . 2 |
31 | 9, 30 | impbida 560 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wb 103 wceq 1284 wcel 1433 class class class wbr 3785 ccnv 4362 cdm 4363 crn 4364 wrel 4368 wfun 4916 wf1 4919 wf1o 4921 cen 6242 cfn 6244 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-coll 3893 ax-sep 3896 ax-nul 3904 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-iinf 4329 |
This theorem depends on definitions: df-bi 115 df-dc 776 df-3or 920 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-ral 2353 df-rex 2354 df-reu 2355 df-rab 2357 df-v 2603 df-sbc 2816 df-csb 2909 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-if 3352 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-iun 3680 df-br 3786 df-opab 3840 df-mpt 3841 df-tr 3876 df-id 4048 df-iord 4121 df-on 4123 df-suc 4126 df-iom 4332 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-f1 4927 df-fo 4928 df-f1o 4929 df-fv 4930 df-1st 5787 df-2nd 5788 df-1o 6024 df-er 6129 df-en 6245 df-fin 6247 |
This theorem is referenced by: f1vrnfibi 6394 |
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