| Metamath Proof Explorer |
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| Description: Function value in an image. (The proof was shortened by Andrew Salmon, 22-Oct-2011.) (An unnecessary distinct variable restriction was removed by David Abernethy, 17-Dec-2011.) |
| Ref | Expression |
|---|---|
| fvelimab |
|
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | elisset 2299 |
. . 3
| |
| 2 | 1 | anim2i 362 |
. 2
|
| 3 | fvex 4689 |
. . . . . 6
| |
| 4 | eleq1 1957 |
. . . . . 6
| |
| 5 | 3, 4 | mpbii 210 |
. . . . 5
|
| 6 | 5 | a1i 8 |
. . . 4
|
| 7 | 6 | r19.23aiv 2211 |
. . 3
|
| 8 | 7 | anim2i 362 |
. 2
|
| 9 | eleq1 1957 |
. . . . . 6
| |
| 10 | eqeq2 1893 |
. . . . . . 7
| |
| 11 | 10 | rexbidv 2124 |
. . . . . 6
|
| 12 | 9, 11 | bibi12d 691 |
. . . . 5
|
| 13 | 12 | imbi2d 674 |
. . . 4
|
| 14 | fnfun 4510 |
. . . . . . 7
| |
| 15 | 14 | adantr 425 |
. . . . . 6
|
| 16 | fndm 4512 |
. . . . . . . 8
| |
| 17 | 16 | sseq2d 2645 |
. . . . . . 7
|
| 18 | 17 | biimpar 461 |
. . . . . 6
|
| 19 | dfimafn 4720 |
. . . . . 6
| |
| 20 | 15, 18, 19 | syl11anc 524 |
. . . . 5
|
| 21 | 20 | abeq2d 2003 |
. . . 4
|
| 22 | 13, 21 | vtoclg 2346 |
. . 3
|
| 23 | 22 | impcom 378 |
. 2
|
| 24 | 2, 8, 23 | pm5.21nd 744 |
1
|
| Colors of variables: wff set class |
| Syntax hints: |
| This theorem is referenced by: ssimaex 4729 ac6sfilem3 5508 pjimai 11748 nocvxmin 14029 npincppr 14501 filnetlem5 15644 raleqfn 15672 f1elima 15719 ismtyhmeolem 15950 heiborlem10 15964 heiborlem11 15965 |
| 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-13 1311 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 ax-un 3790 |
| 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-ral 2109 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 |