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Mirrors > Home > ILE Home > Th. List > foimacnv | Unicode version |
Description: A reverse version of f1imacnv 5352. (Contributed by Jeff Hankins, 16-Jul-2009.) |
Ref | Expression |
---|---|
foimacnv |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | resima 4822 | . 2 | |
2 | fofun 5316 | . . . . . 6 | |
3 | 2 | adantr 274 | . . . . 5 |
4 | funcnvres2 5168 | . . . . 5 | |
5 | 3, 4 | syl 14 | . . . 4 |
6 | 5 | imaeq1d 4850 | . . 3 |
7 | resss 4813 | . . . . . . . . . . 11 | |
8 | cnvss 4682 | . . . . . . . . . . 11 | |
9 | 7, 8 | ax-mp 5 | . . . . . . . . . 10 |
10 | cnvcnvss 4963 | . . . . . . . . . 10 | |
11 | 9, 10 | sstri 3076 | . . . . . . . . 9 |
12 | funss 5112 | . . . . . . . . 9 | |
13 | 11, 2, 12 | mpsyl 65 | . . . . . . . 8 |
14 | 13 | adantr 274 | . . . . . . 7 |
15 | df-ima 4522 | . . . . . . . 8 | |
16 | df-rn 4520 | . . . . . . . 8 | |
17 | 15, 16 | eqtr2i 2139 | . . . . . . 7 |
18 | 14, 17 | jctir 311 | . . . . . 6 |
19 | df-fn 5096 | . . . . . 6 | |
20 | 18, 19 | sylibr 133 | . . . . 5 |
21 | dfdm4 4701 | . . . . . 6 | |
22 | forn 5318 | . . . . . . . . . 10 | |
23 | 22 | sseq2d 3097 | . . . . . . . . 9 |
24 | 23 | biimpar 295 | . . . . . . . 8 |
25 | df-rn 4520 | . . . . . . . 8 | |
26 | 24, 25 | sseqtrdi 3115 | . . . . . . 7 |
27 | ssdmres 4811 | . . . . . . 7 | |
28 | 26, 27 | sylib 121 | . . . . . 6 |
29 | 21, 28 | syl5eqr 2164 | . . . . 5 |
30 | df-fo 5099 | . . . . 5 | |
31 | 20, 29, 30 | sylanbrc 413 | . . . 4 |
32 | foima 5320 | . . . 4 | |
33 | 31, 32 | syl 14 | . . 3 |
34 | 6, 33 | eqtr3d 2152 | . 2 |
35 | 1, 34 | syl5eqr 2164 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1316 wss 3041 ccnv 4508 cdm 4509 crn 4510 cres 4511 cima 4512 wfun 5087 wfn 5088 wfo 5091 |
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 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-br 3900 df-opab 3960 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-fun 5095 df-fn 5096 df-f 5097 df-fo 5099 |
This theorem is referenced by: f1opw2 5944 fopwdom 6698 fisumss 11116 hmeoimaf1o 12394 |
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