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Mirrors > Home > ILE Home > Th. List > f1cnvcnv | Unicode version |
Description: Two ways to express that a set (not necessarily a function) is one-to-one. Each side is equivalent to Definition 6.4(3) of [TakeutiZaring] p. 24, who use the notation "Un2 (A)" for one-to-one. We do not introduce a separate notation since we rarely use it. (Contributed by NM, 13-Aug-2004.) |
Ref | Expression |
---|---|
f1cnvcnv |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-f1 5098 | . 2 | |
2 | dffn2 5244 | . . . 4 | |
3 | dmcnvcnv 4733 | . . . . 5 | |
4 | df-fn 5096 | . . . . 5 | |
5 | 3, 4 | mpbiran2 910 | . . . 4 |
6 | 2, 5 | bitr3i 185 | . . 3 |
7 | relcnv 4887 | . . . . 5 | |
8 | dfrel2 4959 | . . . . 5 | |
9 | 7, 8 | mpbi 144 | . . . 4 |
10 | 9 | funeqi 5114 | . . 3 |
11 | 6, 10 | anbi12ci 456 | . 2 |
12 | 1, 11 | bitri 183 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 103 wb 104 wceq 1316 cvv 2660 ccnv 4508 cdm 4509 wrel 4514 wfun 5087 wfn 5088 wf 5089 wf1 5090 |
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-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 |
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-v 2662 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-br 3900 df-opab 3960 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-fun 5095 df-fn 5096 df-f 5097 df-f1 5098 |
This theorem is referenced by: (None) |
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