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Mirrors > Home > ILE Home > Th. List > respreima | Unicode version |
Description: The preimage of a restricted function. (Contributed by Jeff Madsen, 2-Sep-2009.) |
Ref | Expression |
---|---|
respreima |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funfn 4931 | . . 3 | |
2 | elin 3126 | . . . . . . . . 9 | |
3 | ancom 253 | . . . . . . . . 9 | |
4 | 2, 3 | bitri 173 | . . . . . . . 8 |
5 | 4 | anbi1i 431 | . . . . . . 7 |
6 | fvres 5198 | . . . . . . . . . 10 | |
7 | 6 | eleq1d 2106 | . . . . . . . . 9 |
8 | 7 | adantl 262 | . . . . . . . 8 |
9 | 8 | pm5.32i 427 | . . . . . . 7 |
10 | 5, 9 | bitri 173 | . . . . . 6 |
11 | 10 | a1i 9 | . . . . 5 |
12 | an32 496 | . . . . 5 | |
13 | 11, 12 | syl6bb 185 | . . . 4 |
14 | fnfun 4996 | . . . . . . . 8 | |
15 | funres 4941 | . . . . . . . 8 | |
16 | 14, 15 | syl 14 | . . . . . . 7 |
17 | dmres 4632 | . . . . . . 7 | |
18 | 16, 17 | jctir 296 | . . . . . 6 |
19 | df-fn 4905 | . . . . . 6 | |
20 | 18, 19 | sylibr 137 | . . . . 5 |
21 | elpreima 5286 | . . . . 5 | |
22 | 20, 21 | syl 14 | . . . 4 |
23 | elin 3126 | . . . . 5 | |
24 | elpreima 5286 | . . . . . 6 | |
25 | 24 | anbi1d 438 | . . . . 5 |
26 | 23, 25 | syl5bb 181 | . . . 4 |
27 | 13, 22, 26 | 3bitr4d 209 | . . 3 |
28 | 1, 27 | sylbi 114 | . 2 |
29 | 28 | eqrdv 2038 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wb 98 wceq 1243 wcel 1393 cin 2916 ccnv 4344 cdm 4345 cres 4347 cima 4348 wfun 4896 wfn 4897 cfv 4902 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-sbc 2765 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-br 3765 df-opab 3819 df-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-fv 4910 |
This theorem is referenced by: (None) |
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