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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 5153 | . . 3 | |
2 | elin 3259 | . . . . . . . . 9 | |
3 | ancom 264 | . . . . . . . . 9 | |
4 | 2, 3 | bitri 183 | . . . . . . . 8 |
5 | 4 | anbi1i 453 | . . . . . . 7 |
6 | fvres 5445 | . . . . . . . . . 10 | |
7 | 6 | eleq1d 2208 | . . . . . . . . 9 |
8 | 7 | adantl 275 | . . . . . . . 8 |
9 | 8 | pm5.32i 449 | . . . . . . 7 |
10 | 5, 9 | bitri 183 | . . . . . 6 |
11 | 10 | a1i 9 | . . . . 5 |
12 | an32 551 | . . . . 5 | |
13 | 11, 12 | syl6bb 195 | . . . 4 |
14 | fnfun 5220 | . . . . . . . 8 | |
15 | funres 5164 | . . . . . . . 8 | |
16 | 14, 15 | syl 14 | . . . . . . 7 |
17 | dmres 4840 | . . . . . . 7 | |
18 | 16, 17 | jctir 311 | . . . . . 6 |
19 | df-fn 5126 | . . . . . 6 | |
20 | 18, 19 | sylibr 133 | . . . . 5 |
21 | elpreima 5539 | . . . . 5 | |
22 | 20, 21 | syl 14 | . . . 4 |
23 | elin 3259 | . . . . 5 | |
24 | elpreima 5539 | . . . . . 6 | |
25 | 24 | anbi1d 460 | . . . . 5 |
26 | 23, 25 | syl5bb 191 | . . . 4 |
27 | 13, 22, 26 | 3bitr4d 219 | . . 3 |
28 | 1, 27 | sylbi 120 | . 2 |
29 | 28 | eqrdv 2137 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1331 wcel 1480 cin 3070 ccnv 4538 cdm 4539 cres 4541 cima 4542 wfun 5117 wfn 5118 cfv 5123 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-sbc 2910 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-fv 5131 |
This theorem is referenced by: (None) |
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