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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 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funres 4941 | . . . . 5 | |
2 | funfvex 5192 | . . . . . 6 | |
3 | 2 | ralrimiva 2392 | . . . . 5 |
4 | fnasrng 5343 | . . . . 5 | |
5 | 1, 3, 4 | 3syl 17 | . . . 4 |
6 | 5 | adantr 261 | . . 3 |
7 | 1 | adantr 261 | . . . . 5 |
8 | funfn 4931 | . . . . 5 | |
9 | 7, 8 | sylib 127 | . . . 4 |
10 | dffn5im 5219 | . . . 4 | |
11 | 9, 10 | syl 14 | . . 3 |
12 | imadmrn 4678 | . . . . 5 | |
13 | vex 2560 | . . . . . . . . 9 | |
14 | opexgOLD 3965 | . . . . . . . . 9 | |
15 | 13, 2, 14 | sylancr 393 | . . . . . . . 8 |
16 | 15 | ralrimiva 2392 | . . . . . . 7 |
17 | dmmptg 4818 | . . . . . . 7 | |
18 | 1, 16, 17 | 3syl 17 | . . . . . 6 |
19 | 18 | imaeq2d 4668 | . . . . 5 |
20 | 12, 19 | syl5reqr 2087 | . . . 4 |
21 | 20 | adantr 261 | . . 3 |
22 | 6, 11, 21 | 3eqtr4d 2082 | . 2 |
23 | funmpt 4938 | . . 3 | |
24 | dmresexg 4634 | . . . 4 | |
25 | 24 | adantl 262 | . . 3 |
26 | funimaexg 4983 | . . 3 | |
27 | 23, 25, 26 | sylancr 393 | . 2 |
28 | 22, 27 | eqeltrd 2114 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wceq 1243 wcel 1393 wral 2306 cvv 2557 cop 3378 cmpt 3818 cdm 4345 crn 4346 cres 4347 cima 4348 wfun 4896 wfn 4897 cfv 4902 |
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 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-coll 3872 ax-sep 3875 ax-pow 3927 ax-pr 3944 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-reu 2313 df-rab 2315 df-v 2559 df-sbc 2765 df-csb 2853 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-iun 3659 df-br 3765 df-opab 3819 df-mpt 3820 df-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 df-fv 4910 |
This theorem is referenced by: fnex 5383 ofexg 5716 cofunexg 5738 rdgivallem 5968 frecex 5981 frecsuclem3 5990 |
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