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Mirrors > Home > ILE Home > Th. List > f1orescnv | Unicode version |
Description: The converse of a one-to-one-onto restricted function. (Contributed by Paul Chapman, 21-Apr-2008.) |
Ref | Expression |
---|---|
f1orescnv |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1ocnv 5427 | . . 3 | |
2 | 1 | adantl 275 | . 2 |
3 | funcnvres 5243 | . . . 4 | |
4 | df-ima 4599 | . . . . . 6 | |
5 | dff1o5 5423 | . . . . . . 7 | |
6 | 5 | simprbi 273 | . . . . . 6 |
7 | 4, 6 | syl5eq 2202 | . . . . 5 |
8 | 7 | reseq2d 4866 | . . . 4 |
9 | 3, 8 | sylan9eq 2210 | . . 3 |
10 | f1oeq1 5403 | . . 3 | |
11 | 9, 10 | syl 14 | . 2 |
12 | 2, 11 | mpbid 146 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1335 ccnv 4585 crn 4587 cres 4588 cima 4589 wfun 5164 wf1 5167 wf1o 5169 |
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-res 4598 df-ima 4599 df-fun 5172 df-fn 5173 df-f 5174 df-f1 5175 df-fo 5176 df-f1o 5177 |
This theorem is referenced by: f1oresrab 5632 |
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