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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 4917 | . 2 | |
2 | relres 4847 | . 2 | |
3 | opelf 5294 | . . . . . . 7 | |
4 | 3 | simpld 111 | . . . . . 6 |
5 | 4 | ex 114 | . . . . 5 |
6 | 5 | pm4.71d 390 | . . . 4 |
7 | vex 2689 | . . . . . 6 | |
8 | vex 2689 | . . . . . 6 | |
9 | 7, 8 | opelcnv 4721 | . . . . 5 |
10 | 7 | opelres 4824 | . . . . 5 |
11 | 9, 10 | bitri 183 | . . . 4 |
12 | 6, 11 | syl6bbr 197 | . . 3 |
13 | 3 | simprd 113 | . . . . . 6 |
14 | 13 | ex 114 | . . . . 5 |
15 | 14 | pm4.71d 390 | . . . 4 |
16 | 8 | opelres 4824 | . . . . 5 |
17 | 7, 8 | opelcnv 4721 | . . . . . 6 |
18 | 17 | anbi1i 453 | . . . . 5 |
19 | 16, 18 | bitri 183 | . . . 4 |
20 | 15, 19 | syl6bbr 197 | . . 3 |
21 | 12, 20 | bitr3d 189 | . 2 |
22 | 1, 2, 21 | eqrelrdv 4635 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1331 wcel 1480 cop 3530 ccnv 4538 cres 4541 wf 5119 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-br 3930 df-opab 3990 df-xp 4545 df-rel 4546 df-cnv 4547 df-dm 4549 df-rn 4550 df-res 4551 df-fun 5125 df-fn 5126 df-f 5127 |
This theorem is referenced by: (None) |
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