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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 5202 | . . 3 | |
2 | elin 3291 | . . . . . . . . 9 | |
3 | ancom 264 | . . . . . . . . 9 | |
4 | 2, 3 | bitri 183 | . . . . . . . 8 |
5 | 4 | anbi1i 454 | . . . . . . 7 |
6 | fvres 5494 | . . . . . . . . . 10 | |
7 | 6 | eleq1d 2226 | . . . . . . . . 9 |
8 | 7 | adantl 275 | . . . . . . . 8 |
9 | 8 | pm5.32i 450 | . . . . . . 7 |
10 | 5, 9 | bitri 183 | . . . . . 6 |
11 | 10 | a1i 9 | . . . . 5 |
12 | an32 552 | . . . . 5 | |
13 | 11, 12 | bitrdi 195 | . . . 4 |
14 | fnfun 5269 | . . . . . . . 8 | |
15 | funres 5213 | . . . . . . . 8 | |
16 | 14, 15 | syl 14 | . . . . . . 7 |
17 | dmres 4889 | . . . . . . 7 | |
18 | 16, 17 | jctir 311 | . . . . . 6 |
19 | df-fn 5175 | . . . . . 6 | |
20 | 18, 19 | sylibr 133 | . . . . 5 |
21 | elpreima 5588 | . . . . 5 | |
22 | 20, 21 | syl 14 | . . . 4 |
23 | elin 3291 | . . . . 5 | |
24 | elpreima 5588 | . . . . . 6 | |
25 | 24 | anbi1d 461 | . . . . 5 |
26 | 23, 25 | syl5bb 191 | . . . 4 |
27 | 13, 22, 26 | 3bitr4d 219 | . . 3 |
28 | 1, 27 | sylbi 120 | . 2 |
29 | 28 | eqrdv 2155 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1335 wcel 2128 cin 3101 ccnv 4587 cdm 4588 cres 4590 cima 4591 wfun 5166 wfn 5167 cfv 5172 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-14 2131 ax-ext 2139 ax-sep 4084 ax-pow 4137 ax-pr 4171 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-v 2714 df-sbc 2938 df-un 3106 df-in 3108 df-ss 3115 df-pw 3546 df-sn 3567 df-pr 3568 df-op 3570 df-uni 3775 df-br 3968 df-opab 4028 df-id 4255 df-xp 4594 df-rel 4595 df-cnv 4596 df-co 4597 df-dm 4598 df-rn 4599 df-res 4600 df-ima 4601 df-iota 5137 df-fun 5174 df-fn 5175 df-fv 5180 |
This theorem is referenced by: (None) |
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