| Metamath Proof Explorer |
< Previous
Next >
Related theorems Unicode version |
| Description: Equivalence of function value and binary relation. |
| Ref | Expression |
|---|---|
| fnfvbr.1 |
|
| Ref | Expression |
|---|---|
| fnbrfvb |
|
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | fnfvbr.1 |
. 2
| |
| 2 | eqeq2 1893 |
. . . 4
| |
| 3 | breq2 3342 |
. . . 4
| |
| 4 | 2, 3 | bibi12d 691 |
. . 3
|
| 5 | 4 | imbi2d 674 |
. 2
|
| 6 | fneu 4517 |
. . 3
| |
| 7 | breq1 3341 |
. . . . . . 7
| |
| 8 | 7 | eubidv 1779 |
. . . . . 6
|
| 9 | fveq2 4681 |
. . . . . . . 8
| |
| 10 | 9 | eqeq1d 1892 |
. . . . . . 7
|
| 11 | 10, 7 | bibi12d 691 |
. . . . . 6
|
| 12 | 8, 11 | imbi12d 688 |
. . . . 5
|
| 13 | visset 2295 |
. . . . . 6
| |
| 14 | 13 | tz6.12c 4697 |
. . . . 5
|
| 15 | 12, 14 | vtoclg 2346 |
. . . 4
|
| 16 | 15 | adantl 424 |
. . 3
|
| 17 | 6, 16 | mpd 29 |
. 2
|
| 18 | 1, 5, 17 | vtocl 2339 |
1
|
| Colors of variables: wff set class |
| Syntax hints: |
| This theorem is referenced by: fnopfvb 4713 funbrfvb 4714 fnsnfv 4728 dffo4 4793 dff13 4850 isomin 4876 isoini 4877 1stconst 5070 2ndconst 5071 adjbd1o 11655 bra11 11679 trclss 13935 fvbigcup 14076 |
| This theorem was proved from axioms: ax-1 4 ax-2 5 ax-3 6 ax-mp 7 ax-7 1304 ax-gen 1305 ax-8 1306 ax-9 1307 ax-10 1308 ax-11 1309 ax-12 1310 ax-14 1312 ax-17 1317 ax-4 1319 ax-5o 1321 ax-6o 1324 ax-9o 1481 ax-10o 1500 ax-16 1580 ax-11o 1588 ax-ext 1865 ax-sep 3438 ax-nul 3445 ax-pow 3481 ax-pr 3524 |
| This theorem depends on definitions: df-bi 164 df-or 241 df-an 242 df-ex 1327 df-sb 1536 df-eu 1775 df-mo 1776 df-clab 1872 df-cleq 1877 df-clel 1880 df-ne 2019 df-rex 2110 df-v 2294 df-dif 2597 df-un 2600 df-in 2603 df-ss 2605 df-nul 2876 df-pw 3035 df-sn 3049 df-pr 3050 df-op 3053 df-uni 3178 df-br 3339 df-opab 3396 df-id 3586 df-xp 4000 df-cnv 4002 df-co 4003 df-dm 4004 df-rn 4005 df-res 4006 df-ima 4007 df-fun 4008 df-fn 4009 df-fv 4014 |