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Mirrors > Home > ILE Home > Th. List > f1imacnv | Unicode version |
Description: Preimage of an image. (Contributed by NM, 30-Sep-2004.) |
Ref | Expression |
---|---|
f1imacnv |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | resima 4847 | . 2 | |
2 | df-f1 5123 | . . . . . . 7 | |
3 | 2 | simprbi 273 | . . . . . 6 |
4 | 3 | adantr 274 | . . . . 5 |
5 | funcnvres 5191 | . . . . 5 | |
6 | 4, 5 | syl 14 | . . . 4 |
7 | 6 | imaeq1d 4875 | . . 3 |
8 | f1ores 5375 | . . . . 5 | |
9 | f1ocnv 5373 | . . . . 5 | |
10 | 8, 9 | syl 14 | . . . 4 |
11 | imadmrn 4886 | . . . . 5 | |
12 | f1odm 5364 | . . . . . 6 | |
13 | 12 | imaeq2d 4876 | . . . . 5 |
14 | f1ofo 5367 | . . . . . 6 | |
15 | forn 5343 | . . . . . 6 | |
16 | 14, 15 | syl 14 | . . . . 5 |
17 | 11, 13, 16 | 3eqtr3a 2194 | . . . 4 |
18 | 10, 17 | syl 14 | . . 3 |
19 | 7, 18 | eqtr3d 2172 | . 2 |
20 | 1, 19 | syl5eqr 2184 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1331 wss 3066 ccnv 4533 cdm 4534 crn 4535 cres 4536 cima 4537 wfun 5112 wf 5114 wf1 5115 wfo 5116 wf1o 5117 |
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 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-v 2683 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-br 3925 df-opab 3985 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 |
This theorem is referenced by: f1opw2 5969 ssenen 6738 hmeoopn 12469 hmeocld 12470 hmeontr 12471 |
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