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Mirrors > Home > ILE Home > Th. List > fmpt | Unicode version |
Description: Functionality of the mapping operation. (Contributed by Mario Carneiro, 26-Jul-2013.) (Revised by Mario Carneiro, 31-Aug-2015.) |
Ref | Expression |
---|---|
fmpt.1 |
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Ref | Expression |
---|---|
fmpt |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fmpt.1 |
. . . 4
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2 | 1 | fnmpt 5361 |
. . 3
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3 | 1 | rnmpt 4893 |
. . . 4
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4 | r19.29 2627 |
. . . . . . 7
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5 | eleq1 2252 |
. . . . . . . . 9
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6 | 5 | biimparc 299 |
. . . . . . . 8
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7 | 6 | rexlimivw 2603 |
. . . . . . 7
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8 | 4, 7 | syl 14 |
. . . . . 6
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9 | 8 | ex 115 |
. . . . 5
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10 | 9 | abssdv 3244 |
. . . 4
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11 | 3, 10 | eqsstrid 3216 |
. . 3
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12 | df-f 5239 |
. . 3
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13 | 2, 11, 12 | sylanbrc 417 |
. 2
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14 | fimacnv 5666 |
. . . 4
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15 | 1 | mptpreima 5140 |
. . . 4
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16 | 14, 15 | eqtr3di 2237 |
. . 3
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17 | rabid2 2667 |
. . 3
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18 | 16, 17 | sylib 122 |
. 2
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19 | 13, 18 | impbii 126 |
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 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-rab 2477 df-v 2754 df-sbc 2978 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-br 4019 df-opab 4080 df-mpt 4081 df-id 4311 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-rn 4655 df-res 4656 df-ima 4657 df-iota 5196 df-fun 5237 df-fn 5238 df-f 5239 df-fv 5243 |
This theorem is referenced by: f1ompt 5688 fmpti 5689 fvmptelcdm 5690 fmptd 5691 fmptdf 5694 rnmptss 5698 f1oresrab 5702 idref 5778 f1mpt 5793 f1stres 6185 f2ndres 6186 fmpox 6226 fmpoco 6242 iunon 6310 mptelixpg 6761 dom2lem 6799 uzf 9562 pcmptcl 12377 upxp 14249 txdis1cn 14255 cnmpt11 14260 cnmpt21 14268 fsumcncntop 14533 cncfmpt1f 14561 mulcncflem 14567 mulcncf 14568 cnmptlimc 14620 sincn 14667 coscn 14668 |
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