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Mirrors > Home > ILE Home > Th. List > f1ofveu | Unicode version |
Description: There is one domain element for each value of a one-to-one onto function. (Contributed by NM, 26-May-2006.) |
Ref | Expression |
---|---|
f1ofveu |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1ocnv 5493 |
. . . 4
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2 | f1of 5480 |
. . . 4
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3 | 1, 2 | syl 14 |
. . 3
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4 | feu 5417 |
. . 3
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5 | 3, 4 | sylan 283 |
. 2
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6 | f1ocnvfvb 5802 |
. . . . . 6
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7 | 6 | 3com23 1211 |
. . . . 5
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8 | dff1o4 5488 |
. . . . . . 7
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9 | 8 | simprbi 275 |
. . . . . 6
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10 | fnopfvb 5578 |
. . . . . . 7
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11 | 10 | 3adant3 1019 |
. . . . . 6
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12 | 9, 11 | syl3an1 1282 |
. . . . 5
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13 | 7, 12 | bitrd 188 |
. . . 4
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14 | 13 | 3expa 1205 |
. . 3
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15 | 14 | reubidva 2673 |
. 2
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16 | 5, 15 | mpbird 167 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-reu 2475 df-v 2754 df-sbc 2978 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-br 4019 df-opab 4080 df-id 4311 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-rn 4655 df-res 4656 df-ima 4657 df-iota 5196 df-fun 5237 df-fn 5238 df-f 5239 df-f1 5240 df-fo 5241 df-f1o 5242 df-fv 5243 |
This theorem is referenced by: 1arith2 12403 |
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