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Mirrors > Home > ILE Home > Th. List > funcnv | Unicode version |
Description: The converse of a class is a function iff the class is single-rooted, which means that for any in the range of there is at most one such that . Definition of single-rooted in [Enderton] p. 43. See funcnv2 5230 for a simpler version. (Contributed by NM, 13-Aug-2004.) |
Ref | Expression |
---|---|
funcnv |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 2715 | . . . . . . 7 | |
2 | vex 2715 | . . . . . . 7 | |
3 | 1, 2 | brelrn 4819 | . . . . . 6 |
4 | 3 | pm4.71ri 390 | . . . . 5 |
5 | 4 | mobii 2043 | . . . 4 |
6 | moanimv 2081 | . . . 4 | |
7 | 5, 6 | bitri 183 | . . 3 |
8 | 7 | albii 1450 | . 2 |
9 | funcnv2 5230 | . 2 | |
10 | df-ral 2440 | . 2 | |
11 | 8, 9, 10 | 3bitr4i 211 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wal 1333 wmo 2007 wcel 2128 wral 2435 class class class wbr 3965 ccnv 4585 crn 4587 wfun 5164 |
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 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-14 2131 ax-ext 2139 ax-sep 4082 ax-pow 4135 ax-pr 4169 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-v 2714 df-un 3106 df-in 3108 df-ss 3115 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-br 3966 df-opab 4026 df-id 4253 df-xp 4592 df-rel 4593 df-cnv 4594 df-co 4595 df-dm 4596 df-rn 4597 df-fun 5172 |
This theorem is referenced by: funcnv3 5232 fncnv 5236 |
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