Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 4912 | . 2 | |
2 | relres 4842 | . 2 | |
3 | opelf 5289 | . . . . . . 7 | |
4 | 3 | simpld 111 | . . . . . 6 |
5 | 4 | ex 114 | . . . . 5 |
6 | 5 | pm4.71d 390 | . . . 4 |
7 | vex 2684 | . . . . . 6 | |
8 | vex 2684 | . . . . . 6 | |
9 | 7, 8 | opelcnv 4716 | . . . . 5 |
10 | 7 | opelres 4819 | . . . . 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 4819 | . . . . 5 |
17 | 7, 8 | opelcnv 4716 | . . . . . 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 4630 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1331 wcel 1480 cop 3525 ccnv 4533 cres 4536 wf 5114 |
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 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-v 2683 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-br 3925 df-opab 3985 df-xp 4540 df-rel 4541 df-cnv 4542 df-dm 4544 df-rn 4545 df-res 4546 df-fun 5120 df-fn 5121 df-f 5122 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |