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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 504 | . . . . 5 | |
3 | f1rn 5299 | . . . . . 6 | |
4 | 3 | adantl 275 | . . . . 5 |
5 | f1fn 5300 | . . . . . . . . 9 | |
6 | fnima 5211 | . . . . . . . . 9 | |
7 | 5, 6 | syl 14 | . . . . . . . 8 |
8 | 7 | adantl 275 | . . . . . . 7 |
9 | ssid 3087 | . . . . . . . . 9 | |
10 | 9 | a1i 9 | . . . . . . . 8 |
11 | simpll 503 | . . . . . . . . 9 | |
12 | enfii 6736 | . . . . . . . . 9 | |
13 | 2, 11, 12 | syl2anc 408 | . . . . . . . 8 |
14 | f1imaeng 6654 | . . . . . . . 8 | |
15 | 1, 10, 13, 14 | syl3anc 1201 | . . . . . . 7 |
16 | 8, 15 | eqbrtrrd 3922 | . . . . . 6 |
17 | entr 6646 | . . . . . 6 | |
18 | 16, 11, 17 | syl2anc 408 | . . . . 5 |
19 | fisseneq 6788 | . . . . 5 | |
20 | 2, 4, 18, 19 | syl3anc 1201 | . . . 4 |
21 | dff1o5 5344 | . . . 4 | |
22 | 1, 20, 21 | sylanbrc 413 | . . 3 |
23 | 22 | ex 114 | . 2 |
24 | f1of1 5334 | . 2 | |
25 | 23, 24 | impbid1 141 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1316 wcel 1465 wss 3041 class class class wbr 3899 crn 4510 cima 4512 wfn 5088 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-1o 6281 df-er 6397 df-en 6603 df-fin 6605 |
This theorem is referenced by: iseqf1olemqf1o 10234 crth 11827 pwf1oexmid 13121 |
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