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| Mirrors > Home > ILE Home > Th. List > f1oeng | Unicode version | ||
| Description: The domain and range of a one-to-one, onto function are equinumerous. (Contributed by NM, 19-Jun-1998.) |
| Ref | Expression |
|---|---|
| f1oeng |
|
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | f1ofo 5529 |
. . . 4
| |
| 2 | focdmex 6200 |
. . . 4
| |
| 3 | 1, 2 | syl5 32 |
. . 3
|
| 4 | 3 | imp 124 |
. 2
|
| 5 | f1oen2g 6846 |
. . 3
| |
| 6 | 5 | 3com23 1212 |
. 2
|
| 7 | 4, 6 | mpd3an3 1351 |
1
|
| 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 711 ax-5 1470 ax-7 1471 ax-gen 1472 ax-ie1 1516 ax-ie2 1517 ax-8 1527 ax-10 1528 ax-11 1529 ax-i12 1530 ax-bndl 1532 ax-4 1533 ax-17 1549 ax-i9 1553 ax-ial 1557 ax-i5r 1558 ax-13 2178 ax-14 2179 ax-ext 2187 ax-coll 4159 ax-sep 4162 ax-pow 4218 ax-pr 4253 ax-un 4480 |
| This theorem depends on definitions: df-bi 117 df-3an 983 df-tru 1376 df-nf 1484 df-sb 1786 df-eu 2057 df-mo 2058 df-clab 2192 df-cleq 2198 df-clel 2201 df-nfc 2337 df-ral 2489 df-rex 2490 df-reu 2491 df-rab 2493 df-v 2774 df-sbc 2999 df-csb 3094 df-un 3170 df-in 3172 df-ss 3179 df-pw 3618 df-sn 3639 df-pr 3640 df-op 3642 df-uni 3851 df-iun 3929 df-br 4045 df-opab 4106 df-mpt 4107 df-id 4340 df-xp 4681 df-rel 4682 df-cnv 4683 df-co 4684 df-dm 4685 df-rn 4686 df-res 4687 df-ima 4688 df-iota 5232 df-fun 5273 df-fn 5274 df-f 5275 df-f1 5276 df-fo 5277 df-f1o 5278 df-fv 5279 df-en 6828 |
| This theorem is referenced by: f1oen 6850 f1imaeng 6884 xpen 6942 fidifsnen 6967 dif1en 6976 f1ofi 7045 f1dmvrnfibi 7046 omp1eom 7197 endjusym 7198 eninl 7199 eninr 7200 summodclem2 11693 zsumdc 11695 prodmodclem2 11888 zproddc 11890 eulerthlemh 12553 4sqlem11 12724 ssnnctlemct 12817 conjsubgen 13614 znfi 14417 znhash 14418 2omapen 15933 pwf1oexmid 15936 |
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