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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 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | resima 4940 |
. 2
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2 | df-f1 5221 |
. . . . . . 7
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3 | 2 | simprbi 275 |
. . . . . 6
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4 | 3 | adantr 276 |
. . . . 5
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5 | funcnvres 5289 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
6 | 4, 5 | syl 14 |
. . . 4
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7 | 6 | imaeq1d 4969 |
. . 3
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8 | f1ores 5476 |
. . . . 5
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9 | f1ocnv 5474 |
. . . . 5
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10 | 8, 9 | syl 14 |
. . . 4
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11 | imadmrn 4980 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
12 | f1odm 5465 |
. . . . . 6
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13 | 12 | imaeq2d 4970 |
. . . . 5
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14 | f1ofo 5468 |
. . . . . 6
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15 | forn 5441 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
16 | 14, 15 | syl 14 |
. . . . 5
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17 | 11, 13, 16 | 3eqtr3a 2234 |
. . . 4
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18 | 10, 17 | syl 14 |
. . 3
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19 | 7, 18 | eqtr3d 2212 |
. 2
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20 | 1, 19 | eqtr3id 2224 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4121 ax-pow 4174 ax-pr 4209 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2739 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-br 4004 df-opab 4065 df-id 4293 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-rn 4637 df-res 4638 df-ima 4639 df-fun 5218 df-fn 5219 df-f 5220 df-f1 5221 df-fo 5222 df-f1o 5223 |
This theorem is referenced by: f1opw2 6076 ssenen 6850 hmeoopn 13747 hmeocld 13748 hmeontr 13749 |
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