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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 4951 | . . 3 | |
2 | elin 3155 | . . . . . . . . 9 | |
3 | ancom 262 | . . . . . . . . 9 | |
4 | 2, 3 | bitri 182 | . . . . . . . 8 |
5 | 4 | anbi1i 445 | . . . . . . 7 |
6 | fvres 5219 | . . . . . . . . . 10 | |
7 | 6 | eleq1d 2147 | . . . . . . . . 9 |
8 | 7 | adantl 271 | . . . . . . . 8 |
9 | 8 | pm5.32i 441 | . . . . . . 7 |
10 | 5, 9 | bitri 182 | . . . . . 6 |
11 | 10 | a1i 9 | . . . . 5 |
12 | an32 526 | . . . . 5 | |
13 | 11, 12 | syl6bb 194 | . . . 4 |
14 | fnfun 5016 | . . . . . . . 8 | |
15 | funres 4961 | . . . . . . . 8 | |
16 | 14, 15 | syl 14 | . . . . . . 7 |
17 | dmres 4650 | . . . . . . 7 | |
18 | 16, 17 | jctir 306 | . . . . . 6 |
19 | df-fn 4925 | . . . . . 6 | |
20 | 18, 19 | sylibr 132 | . . . . 5 |
21 | elpreima 5307 | . . . . 5 | |
22 | 20, 21 | syl 14 | . . . 4 |
23 | elin 3155 | . . . . 5 | |
24 | elpreima 5307 | . . . . . 6 | |
25 | 24 | anbi1d 452 | . . . . 5 |
26 | 23, 25 | syl5bb 190 | . . . 4 |
27 | 13, 22, 26 | 3bitr4d 218 | . . 3 |
28 | 1, 27 | sylbi 119 | . 2 |
29 | 28 | eqrdv 2079 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wb 103 wceq 1284 wcel 1433 cin 2972 ccnv 4362 cdm 4363 cres 4365 cima 4366 wfun 4916 wfn 4917 cfv 4922 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-v 2603 df-sbc 2816 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-opab 3840 df-id 4048 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-fv 4930 |
This theorem is referenced by: (None) |
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