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Mirrors > Home > ILE Home > Th. List > fcnvres | Unicode version |
Description: The converse of a restriction of a function. (Contributed by NM, 26-Mar-1998.) |
Ref | Expression |
---|---|
fcnvres |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relcnv 4703 | . 2 | |
2 | relres 4639 | . 2 | |
3 | opelf 5062 | . . . . . . 7 | |
4 | 3 | simpld 105 | . . . . . 6 |
5 | 4 | ex 108 | . . . . 5 |
6 | 5 | pm4.71d 373 | . . . 4 |
7 | vex 2560 | . . . . . 6 | |
8 | vex 2560 | . . . . . 6 | |
9 | 7, 8 | opelcnv 4517 | . . . . 5 |
10 | 7 | opelres 4617 | . . . . 5 |
11 | 9, 10 | bitri 173 | . . . 4 |
12 | 6, 11 | syl6bbr 187 | . . 3 |
13 | 3 | simprd 107 | . . . . . 6 |
14 | 13 | ex 108 | . . . . 5 |
15 | 14 | pm4.71d 373 | . . . 4 |
16 | 8 | opelres 4617 | . . . . 5 |
17 | 7, 8 | opelcnv 4517 | . . . . . 6 |
18 | 17 | anbi1i 431 | . . . . 5 |
19 | 16, 18 | bitri 173 | . . . 4 |
20 | 15, 19 | syl6bbr 187 | . . 3 |
21 | 12, 20 | bitr3d 179 | . 2 |
22 | 1, 2, 21 | eqrelrdv 4436 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wceq 1243 wcel 1393 cop 3378 ccnv 4344 cres 4347 wf 4898 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-br 3765 df-opab 3819 df-xp 4351 df-rel 4352 df-cnv 4353 df-dm 4355 df-rn 4356 df-res 4357 df-fun 4904 df-fn 4905 df-f 4906 |
This theorem is referenced by: (None) |
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