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Mirrors > Home > ILE Home > Th. List > resfunexg | Unicode version |
Description: The restriction of a function to a set exists. Compare Proposition 6.17 of [TakeutiZaring] p. 28. (Contributed by NM, 7-Apr-1995.) (Revised by Mario Carneiro, 22-Jun-2013.) |
Ref | Expression |
---|---|
resfunexg |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funres 4884 |
. . . . 5
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2 | funfvex 5135 |
. . . . . 6
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3 | 2 | ralrimiva 2386 |
. . . . 5
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4 | fnasrng 5286 |
. . . . 5
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5 | 1, 3, 4 | 3syl 17 |
. . . 4
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6 | 5 | adantr 261 |
. . 3
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7 | 1 | adantr 261 |
. . . . 5
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8 | funfn 4874 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
9 | 7, 8 | sylib 127 |
. . . 4
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10 | dffn5im 5162 |
. . . 4
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11 | 9, 10 | syl 14 |
. . 3
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12 | imadmrn 4621 |
. . . . 5
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13 | vex 2554 |
. . . . . . . . 9
![]() ![]() ![]() ![]() | |
14 | opexgOLD 3956 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
15 | 13, 2, 14 | sylancr 393 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
16 | 15 | ralrimiva 2386 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
17 | dmmptg 4761 |
. . . . . . 7
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18 | 1, 16, 17 | 3syl 17 |
. . . . . 6
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19 | 18 | imaeq2d 4611 |
. . . . 5
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20 | 12, 19 | syl5reqr 2084 |
. . . 4
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21 | 20 | adantr 261 |
. . 3
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22 | 6, 11, 21 | 3eqtr4d 2079 |
. 2
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23 | funmpt 4881 |
. . 3
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24 | dmresexg 4577 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
25 | 24 | adantl 262 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
26 | funimaexg 4926 |
. . 3
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27 | 23, 25, 26 | sylancr 393 |
. 2
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28 | 22, 27 | eqeltrd 2111 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-coll 3863 ax-sep 3866 ax-pow 3918 ax-pr 3935 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-rex 2306 df-reu 2307 df-rab 2309 df-v 2553 df-sbc 2759 df-csb 2847 df-un 2916 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-uni 3572 df-iun 3650 df-br 3756 df-opab 3810 df-mpt 3811 df-id 4021 df-xp 4294 df-rel 4295 df-cnv 4296 df-co 4297 df-dm 4298 df-rn 4299 df-res 4300 df-ima 4301 df-iota 4810 df-fun 4847 df-fn 4848 df-f 4849 df-f1 4850 df-fo 4851 df-f1o 4852 df-fv 4853 |
This theorem is referenced by: fnex 5326 ofexg 5658 cofunexg 5680 rdgivallem 5908 frecsuclem3 5929 |
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