| 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 1521 |
. . . 4
| |
| 3 | breq2 2673 |
. . . 4
| |
| 4 | 2, 3 | bibi12d 631 |
. . 3
|
| 5 | 4 | imbi2d 614 |
. 2
|
| 6 | fneu 3667 |
. . 3
| |
| 7 | breq1 2672 |
. . . . . . 7
| |
| 8 | 7 | eubidv 1419 |
. . . . . 6
|
| 9 | fveq2 3800 |
. . . . . . . 8
| |
| 10 | 9 | eqeq1d 1520 |
. . . . . . 7
|
| 11 | 10, 7 | bibi12d 631 |
. . . . . 6
|
| 12 | 8, 11 | imbi12d 628 |
. . . . 5
|
| 13 | visset 1851 |
. . . . . 6
| |
| 14 | 13 | tz6.12c 3816 |
. . . . 5
|
| 15 | 12, 14 | vtoclg 1885 |
. . . 4
|
| 16 | 15 | adantl 388 |
. . 3
|
| 17 | 6, 16 | mpd 26 |
. 2
|
| 18 | 1, 5, 17 | vtocl 1880 |
1
|
| Colors of variables: wff set class |
| Syntax hints: |
| This theorem is referenced by: fnopfvb 3830 funbrfvb 3831 fnsnfv 3843 dffo4 3896 dff13 3950 isomin 3975 isoini 3976 1stconst 4206 2ndconst 4207 adjbd1o 10135 bra11 10158 |
| This theorem was proved from axioms: ax-1 4 ax-2 5 ax-3 6 ax-mp 7 ax-7 994 ax-gen 995 ax-8 996 ax-10 998 ax-11 999 ax-12 1000 ax-13 1001 ax-14 1002 ax-17 1003 ax-4 1005 ax-5o 1007 ax-6o 1010 ax-9o 1155 ax-10o 1173 ax-16 1243 ax-11o 1251 ax-ext 1494 ax-sep 2754 ax-pow 2794 ax-pr 2832 |
| This theorem depends on definitions: df-bi 145 df-or 222 df-an 223 df-ex 1013 df-sb 1205 df-eu 1415 df-mo 1416 df-clab 1500 df-cleq 1505 df-clel 1508 df-ne 1624 df-rex 1688 df-v 1850 df-dif 2093 df-un 2094 df-in 2095 df-ss 2097 df-nul 2325 df-pw 2447 df-sn 2457 df-pr 2458 df-op 2461 df-uni 2552 df-br 2670 df-opab 2718 df-id 2889 df-xp 3239 df-cnv 3241 df-co 3242 df-dm 3243 df-rn 3244 df-res 3245 df-ima 3246 df-fun 3247 df-fn 3248 df-fv 3253 |