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Mirrors > Home > ILE Home > Th. List > restval | Unicode version |
Description: The subspace topology induced by the topology on the set . (Contributed by FL, 20-Sep-2010.) (Revised by Mario Carneiro, 1-May-2015.) |
Ref | Expression |
---|---|
restval | ↾t |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 2741 | . 2 | |
2 | elex 2741 | . 2 | |
3 | mptexg 5718 | . . . . 5 | |
4 | rnexg 4874 | . . . . 5 | |
5 | 3, 4 | syl 14 | . . . 4 |
6 | 5 | adantr 274 | . . 3 |
7 | simpl 108 | . . . . . 6 | |
8 | simpr 109 | . . . . . . 7 | |
9 | 8 | ineq2d 3328 | . . . . . 6 |
10 | 7, 9 | mpteq12dv 4069 | . . . . 5 |
11 | 10 | rneqd 4838 | . . . 4 |
12 | df-rest 12567 | . . . 4 ↾t | |
13 | 11, 12 | ovmpoga 5979 | . . 3 ↾t |
14 | 6, 13 | mpd3an3 1333 | . 2 ↾t |
15 | 1, 2, 14 | syl2an 287 | 1 ↾t |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1348 wcel 2141 cvv 2730 cin 3120 cmpt 4048 crn 4610 (class class class)co 5850 ↾t crest 12565 |
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-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4102 ax-sep 4105 ax-pow 4158 ax-pr 4192 ax-un 4416 ax-setind 4519 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-iun 3873 df-br 3988 df-opab 4049 df-mpt 4050 df-id 4276 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-rn 4620 df-res 4621 df-ima 4622 df-iota 5158 df-fun 5198 df-fn 5199 df-f 5200 df-f1 5201 df-fo 5202 df-f1o 5203 df-fv 5204 df-ov 5853 df-oprab 5854 df-mpo 5855 df-rest 12567 |
This theorem is referenced by: elrest 12572 restid2 12574 tgrest 12884 resttopon 12886 restco 12889 rest0 12894 |
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