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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 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1ocnv 5453 | . . . 4 | |
2 | f1of 5440 | . . . 4 | |
3 | 1, 2 | syl 14 | . . 3 |
4 | feu 5378 | . . 3 | |
5 | 3, 4 | sylan 281 | . 2 |
6 | f1ocnvfvb 5757 | . . . . . 6 | |
7 | 6 | 3com23 1204 | . . . . 5 |
8 | dff1o4 5448 | . . . . . . 7 | |
9 | 8 | simprbi 273 | . . . . . 6 |
10 | fnopfvb 5536 | . . . . . . 7 | |
11 | 10 | 3adant3 1012 | . . . . . 6 |
12 | 9, 11 | syl3an1 1266 | . . . . 5 |
13 | 7, 12 | bitrd 187 | . . . 4 |
14 | 13 | 3expa 1198 | . . 3 |
15 | 14 | reubidva 2652 | . 2 |
16 | 5, 15 | mpbird 166 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 973 wceq 1348 wcel 2141 wreu 2450 cop 3584 ccnv 4608 wfn 5191 wf 5192 wf1o 5195 cfv 5196 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-pow 4158 ax-pr 4192 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-reu 2455 df-v 2732 df-sbc 2956 df-un 3125 df-in 3127 df-ss 3134 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-br 3988 df-opab 4049 df-id 4276 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-rn 4620 df-res 4621 df-ima 4622 df-iota 5158 df-fun 5198 df-fn 5199 df-f 5200 df-f1 5201 df-fo 5202 df-f1o 5203 df-fv 5204 |
This theorem is referenced by: 1arith2 12313 |
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