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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 5172 |
. . . . 5
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2 | funfvex 5446 |
. . . . . 6
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3 | 2 | ralrimiva 2508 |
. . . . 5
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4 | fnasrng 5608 |
. . . . 5
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5 | 1, 3, 4 | 3syl 17 |
. . . 4
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6 | 5 | adantr 274 |
. . 3
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7 | 1 | adantr 274 |
. . . . 5
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8 | funfn 5161 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
9 | 7, 8 | sylib 121 |
. . . 4
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10 | dffn5im 5475 |
. . . 4
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11 | 9, 10 | syl 14 |
. . 3
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12 | imadmrn 4899 |
. . . . 5
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13 | vex 2692 |
. . . . . . . . 9
![]() ![]() ![]() ![]() | |
14 | opexg 4158 |
. . . . . . . . 9
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15 | 13, 2, 14 | sylancr 411 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
16 | 15 | ralrimiva 2508 |
. . . . . . 7
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17 | dmmptg 5044 |
. . . . . . 7
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18 | 1, 16, 17 | 3syl 17 |
. . . . . 6
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19 | 18 | imaeq2d 4889 |
. . . . 5
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20 | 12, 19 | syl5reqr 2188 |
. . . 4
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21 | 20 | adantr 274 |
. . 3
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22 | 6, 11, 21 | 3eqtr4d 2183 |
. 2
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23 | funmpt 5169 |
. . 3
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24 | dmresexg 4850 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
25 | 24 | adantl 275 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
26 | funimaexg 5215 |
. . 3
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27 | 23, 25, 26 | sylancr 411 |
. 2
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28 | 22, 27 | eqeltrd 2217 |
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 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-coll 4051 ax-sep 4054 ax-pow 4106 ax-pr 4139 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-reu 2424 df-rab 2426 df-v 2691 df-sbc 2914 df-csb 3008 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-iun 3823 df-br 3938 df-opab 3998 df-mpt 3999 df-id 4223 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 |
This theorem is referenced by: fnex 5650 ofexg 5994 cofunexg 6017 rdgivallem 6286 frecex 6299 frecsuclem 6311 djudoml 7092 djudomr 7093 fihashf1rn 10567 qnnen 11980 |
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