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Mirrors > Home > ILE Home > Th. List > preimaf1ofi | Unicode version |
Description: The preimage of a finite set under a one-to-one, onto function is finite. (Contributed by Jim Kingdon, 24-Sep-2022.) |
Ref | Expression |
---|---|
preimaf1ofi.ss | |
preimaf1ofi.f | |
preimaf1ofi.c |
Ref | Expression |
---|---|
preimaf1ofi |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | preimaf1ofi.c | . 2 | |
2 | preimaf1ofi.f | . . . 4 | |
3 | f1ocnv 5348 | . . . 4 | |
4 | f1of1 5334 | . . . 4 | |
5 | 2, 3, 4 | 3syl 17 | . . 3 |
6 | preimaf1ofi.ss | . . 3 | |
7 | f1ores 5350 | . . 3 | |
8 | 5, 6, 7 | syl2anc 408 | . 2 |
9 | f1ofi 6799 | . 2 | |
10 | 1, 8, 9 | syl2anc 408 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wcel 1465 wss 3041 ccnv 4508 cres 4511 cima 4512 wf1 5090 wf1o 5092 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-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-pow 4068 ax-pr 4101 ax-un 4325 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 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-ral 2398 df-rex 2399 df-reu 2400 df-rab 2402 df-v 2662 df-sbc 2883 df-csb 2976 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-iun 3785 df-br 3900 df-opab 3960 df-mpt 3961 df-id 4185 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-er 6397 df-en 6603 df-fin 6605 |
This theorem is referenced by: fisumss 11129 |
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