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Mirrors > Home > MPE Home > Th. List > f1dmvrnfibi | Structured version Visualization version 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 8251. (Contributed by AV, 10-Jan-2020.) |
Ref | Expression |
---|---|
f1dmvrnfibi |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rnfi 8249 | . 2 | |
2 | simpr 477 | . . . . 5 | |
3 | f1dm 6105 | . . . . . . . . 9 | |
4 | f1f1orn 6148 | . . . . . . . . 9 | |
5 | eleq1 2689 | . . . . . . . . . . . . 13 | |
6 | f1oeq2 6128 | . . . . . . . . . . . . 13 | |
7 | 5, 6 | anbi12d 747 | . . . . . . . . . . . 12 |
8 | 7 | eqcoms 2630 | . . . . . . . . . . 11 |
9 | 8 | biimpd 219 | . . . . . . . . . 10 |
10 | 9 | expcomd 454 | . . . . . . . . 9 |
11 | 3, 4, 10 | sylc 65 | . . . . . . . 8 |
12 | 11 | impcom 446 | . . . . . . 7 |
13 | 12 | adantr 481 | . . . . . 6 |
14 | f1oeng 7974 | . . . . . 6 | |
15 | 13, 14 | syl 17 | . . . . 5 |
16 | enfii 8177 | . . . . 5 | |
17 | 2, 15, 16 | syl2anc 693 | . . . 4 |
18 | f1fun 6103 | . . . . . 6 | |
19 | 18 | ad2antlr 763 | . . . . 5 |
20 | fundmfibi 8245 | . . . . 5 | |
21 | 19, 20 | syl 17 | . . . 4 |
22 | 17, 21 | mpbird 247 | . . 3 |
23 | 22 | ex 450 | . 2 |
24 | 1, 23 | impbid2 216 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 wceq 1483 wcel 1990 class class class wbr 4653 cdm 5114 crn 5115 wfun 5882 wf1 5885 wf1o 5887 cen 7952 cfn 7955 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-en 7956 df-dom 7957 df-fin 7959 |
This theorem is referenced by: f1vrnfibi 8251 fmtnoinf 41448 |
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