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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 4976 | . 2 | |
2 | relres 4906 | . 2 | |
3 | opelf 5353 | . . . . . . 7 | |
4 | 3 | simpld 111 | . . . . . 6 |
5 | 4 | ex 114 | . . . . 5 |
6 | 5 | pm4.71d 391 | . . . 4 |
7 | vex 2724 | . . . . . 6 | |
8 | vex 2724 | . . . . . 6 | |
9 | 7, 8 | opelcnv 4780 | . . . . 5 |
10 | 7 | opelres 4883 | . . . . 5 |
11 | 9, 10 | bitri 183 | . . . 4 |
12 | 6, 11 | bitr4di 197 | . . 3 |
13 | 3 | simprd 113 | . . . . . 6 |
14 | 13 | ex 114 | . . . . 5 |
15 | 14 | pm4.71d 391 | . . . 4 |
16 | 8 | opelres 4883 | . . . . 5 |
17 | 7, 8 | opelcnv 4780 | . . . . . 6 |
18 | 17 | anbi1i 454 | . . . . 5 |
19 | 16, 18 | bitri 183 | . . . 4 |
20 | 15, 19 | bitr4di 197 | . . 3 |
21 | 12, 20 | bitr3d 189 | . 2 |
22 | 1, 2, 21 | eqrelrdv 4694 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1342 wcel 2135 cop 3573 ccnv 4597 cres 4600 wf 5178 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-v 2723 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-br 3977 df-opab 4038 df-xp 4604 df-rel 4605 df-cnv 4606 df-dm 4608 df-rn 4609 df-res 4610 df-fun 5184 df-fn 5185 df-f 5186 |
This theorem is referenced by: (None) |
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