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Mirrors > Home > MPE Home > Th. List > fnressn | Structured version Visualization version Unicode version |
Description: A function restricted to a singleton. (Contributed by NM, 9-Oct-2004.) |
Ref | Expression |
---|---|
fnressn |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sneq 3980 |
. . . . . 6
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2 | 1 | reseq2d 5108 |
. . . . 5
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3 | fveq2 5870 |
. . . . . . 7
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4 | opeq12 4171 |
. . . . . . 7
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5 | 3, 4 | mpdan 675 |
. . . . . 6
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6 | 5 | sneqd 3982 |
. . . . 5
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7 | 2, 6 | eqeq12d 2468 |
. . . 4
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8 | 7 | imbi2d 318 |
. . 3
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9 | vex 3050 |
. . . . . . 7
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10 | 9 | snss 4099 |
. . . . . 6
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11 | fnssres 5694 |
. . . . . 6
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12 | 10, 11 | sylan2b 478 |
. . . . 5
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13 | dffn2 5735 |
. . . . . 6
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14 | 9 | fsn2 6067 |
. . . . . 6
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15 | fvex 5880 |
. . . . . . . 8
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16 | 15 | biantrur 509 |
. . . . . . 7
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17 | ssnid 3999 |
. . . . . . . . . . 11
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18 | fvres 5884 |
. . . . . . . . . . 11
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19 | 17, 18 | ax-mp 5 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
20 | 19 | opeq2i 4173 |
. . . . . . . . 9
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21 | 20 | sneqi 3981 |
. . . . . . . 8
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22 | 21 | eqeq2i 2465 |
. . . . . . 7
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23 | 16, 22 | bitr3i 255 |
. . . . . 6
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24 | 13, 14, 23 | 3bitri 275 |
. . . . 5
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25 | 12, 24 | sylib 200 |
. . . 4
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26 | 25 | expcom 437 |
. . 3
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27 | 8, 26 | vtoclga 3115 |
. 2
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28 | 27 | impcom 432 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1671 ax-4 1684 ax-5 1760 ax-6 1807 ax-7 1853 ax-9 1898 ax-10 1917 ax-11 1922 ax-12 1935 ax-13 2093 ax-ext 2433 ax-sep 4528 ax-nul 4537 ax-pr 4642 |
This theorem depends on definitions: df-bi 189 df-or 372 df-an 373 df-3an 988 df-tru 1449 df-ex 1666 df-nf 1670 df-sb 1800 df-eu 2305 df-mo 2306 df-clab 2440 df-cleq 2446 df-clel 2449 df-nfc 2583 df-ne 2626 df-ral 2744 df-rex 2745 df-reu 2746 df-rab 2748 df-v 3049 df-sbc 3270 df-dif 3409 df-un 3411 df-in 3413 df-ss 3420 df-nul 3734 df-if 3884 df-sn 3971 df-pr 3973 df-op 3977 df-uni 4202 df-br 4406 df-opab 4465 df-id 4752 df-xp 4843 df-rel 4844 df-cnv 4845 df-co 4846 df-dm 4847 df-rn 4848 df-res 4849 df-ima 4850 df-iota 5549 df-fun 5587 df-fn 5588 df-f 5589 df-f1 5590 df-fo 5591 df-f1o 5592 df-fv 5593 |
This theorem is referenced by: funressn 6082 fressnfv 6083 fnsnsplit 6106 canthp1lem2 9083 fseq1p1m1 11875 dprd2da 17687 dmdprdpr 17694 dprdpr 17695 dpjlem 17696 pgpfaclem1 17726 islindf4 19408 xpstopnlem1 20836 ptcmpfi 20840 2pthlem1 25337 eupath2lem3 25719 ginvsn 26089 subfacp1lem5 29919 cvmliftlem10 30029 poimirlem9 31961 2pthdlem1 39755 |
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