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Mirrors > Home > ILE Home > Th. List > f1vrnfibi | Unicode version |
Description: A one-to-one function which is a set is finite if and only if its range is finite. See also f1dmvrnfibi 6643. (Contributed by AV, 10-Jan-2020.) |
Ref | Expression |
---|---|
f1vrnfibi |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1dm 5215 |
. . . 4
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2 | dmexg 4693 |
. . . . 5
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3 | eleq1 2150 |
. . . . . 6
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4 | 3 | eqcoms 2091 |
. . . . 5
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5 | 2, 4 | syl5ibr 154 |
. . . 4
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6 | 1, 5 | syl 14 |
. . 3
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7 | 6 | impcom 123 |
. 2
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8 | f1dmvrnfibi 6643 |
. 2
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9 | 7, 8 | sylancom 411 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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 579 ax-in2 580 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-13 1449 ax-14 1450 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 ax-coll 3952 ax-sep 3955 ax-nul 3963 ax-pow 4007 ax-pr 4034 ax-un 4258 ax-setind 4351 ax-iinf 4401 |
This theorem depends on definitions: df-bi 115 df-dc 781 df-3or 925 df-3an 926 df-tru 1292 df-fal 1295 df-nf 1395 df-sb 1693 df-eu 1951 df-mo 1952 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-ne 2256 df-ral 2364 df-rex 2365 df-reu 2366 df-rab 2368 df-v 2621 df-sbc 2841 df-csb 2934 df-dif 3001 df-un 3003 df-in 3005 df-ss 3012 df-nul 3287 df-if 3392 df-pw 3429 df-sn 3450 df-pr 3451 df-op 3453 df-uni 3652 df-int 3687 df-iun 3730 df-br 3844 df-opab 3898 df-mpt 3899 df-tr 3935 df-id 4118 df-iord 4191 df-on 4193 df-suc 4196 df-iom 4404 df-xp 4442 df-rel 4443 df-cnv 4444 df-co 4445 df-dm 4446 df-rn 4447 df-res 4448 df-ima 4449 df-iota 4975 df-fun 5012 df-fn 5013 df-f 5014 df-f1 5015 df-fo 5016 df-f1o 5017 df-fv 5018 df-1st 5903 df-2nd 5904 df-1o 6173 df-er 6282 df-en 6448 df-fin 6450 |
This theorem is referenced by: negfi 10646 |
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