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Mirrors > Home > ILE Home > Th. List > f1finf1o | Unicode version |
Description: Any injection from one finite set to another of equal size must be a bijection. (Contributed by Jeff Madsen, 5-Jun-2010.) |
Ref | Expression |
---|---|
f1finf1o |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 109 | . . . 4 | |
2 | simplr 519 | . . . . 5 | |
3 | f1rn 5329 | . . . . . 6 | |
4 | 3 | adantl 275 | . . . . 5 |
5 | f1fn 5330 | . . . . . . . . 9 | |
6 | fnima 5241 | . . . . . . . . 9 | |
7 | 5, 6 | syl 14 | . . . . . . . 8 |
8 | 7 | adantl 275 | . . . . . . 7 |
9 | ssid 3117 | . . . . . . . . 9 | |
10 | 9 | a1i 9 | . . . . . . . 8 |
11 | simpll 518 | . . . . . . . . 9 | |
12 | enfii 6768 | . . . . . . . . 9 | |
13 | 2, 11, 12 | syl2anc 408 | . . . . . . . 8 |
14 | f1imaeng 6686 | . . . . . . . 8 | |
15 | 1, 10, 13, 14 | syl3anc 1216 | . . . . . . 7 |
16 | 8, 15 | eqbrtrrd 3952 | . . . . . 6 |
17 | entr 6678 | . . . . . 6 | |
18 | 16, 11, 17 | syl2anc 408 | . . . . 5 |
19 | fisseneq 6820 | . . . . 5 | |
20 | 2, 4, 18, 19 | syl3anc 1216 | . . . 4 |
21 | dff1o5 5376 | . . . 4 | |
22 | 1, 20, 21 | sylanbrc 413 | . . 3 |
23 | 22 | ex 114 | . 2 |
24 | f1of1 5366 | . 2 | |
25 | 23, 24 | impbid1 141 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1331 wcel 1480 wss 3071 class class class wbr 3929 crn 4540 cima 4542 wfn 5118 wf1 5120 wf1o 5122 cen 6632 cfn 6634 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-if 3475 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-iord 4288 df-on 4290 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-1o 6313 df-er 6429 df-en 6635 df-fin 6637 |
This theorem is referenced by: iseqf1olemqf1o 10266 crth 11900 pwf1oexmid 13194 |
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