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Mirrors > Home > ILE Home > Th. List > restsspw | Unicode version |
Description: The subspace topology is a collection of subsets of the restriction set. (Contributed by Mario Carneiro, 13-Aug-2015.) |
Ref | Expression |
---|---|
restsspw | ↾t |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rest 12558 | . . . . . . 7 ↾t | |
2 | 1 | elmpocl 6036 | . . . . . 6 ↾t |
3 | elrest 12563 | . . . . . 6 ↾t | |
4 | 2, 3 | syl 14 | . . . . 5 ↾t ↾t |
5 | 4 | ibi 175 | . . . 4 ↾t |
6 | inss2 3343 | . . . . . 6 | |
7 | sseq1 3165 | . . . . . 6 | |
8 | 6, 7 | mpbiri 167 | . . . . 5 |
9 | 8 | rexlimivw 2579 | . . . 4 |
10 | 5, 9 | syl 14 | . . 3 ↾t |
11 | velpw 3566 | . . 3 | |
12 | 10, 11 | sylibr 133 | . 2 ↾t |
13 | 12 | ssriv 3146 | 1 ↾t |
Colors of variables: wff set class |
Syntax hints: wa 103 wb 104 wceq 1343 wcel 2136 wrex 2445 cvv 2726 cin 3115 wss 3116 cpw 3559 cmpt 4043 crn 4605 (class class class)co 5842 ↾t crest 12556 |
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 604 ax-in2 605 ax-io 699 ax-5 1435 ax-7 1436 ax-gen 1437 ax-ie1 1481 ax-ie2 1482 ax-8 1492 ax-10 1493 ax-11 1494 ax-i12 1495 ax-bndl 1497 ax-4 1498 ax-17 1514 ax-i9 1518 ax-ial 1522 ax-i5r 1523 ax-13 2138 ax-14 2139 ax-ext 2147 ax-coll 4097 ax-sep 4100 ax-pow 4153 ax-pr 4187 ax-un 4411 ax-setind 4514 |
This theorem depends on definitions: df-bi 116 df-3an 970 df-tru 1346 df-fal 1349 df-nf 1449 df-sb 1751 df-eu 2017 df-mo 2018 df-clab 2152 df-cleq 2158 df-clel 2161 df-nfc 2297 df-ne 2337 df-ral 2449 df-rex 2450 df-reu 2451 df-rab 2453 df-v 2728 df-sbc 2952 df-csb 3046 df-dif 3118 df-un 3120 df-in 3122 df-ss 3129 df-pw 3561 df-sn 3582 df-pr 3583 df-op 3585 df-uni 3790 df-iun 3868 df-br 3983 df-opab 4044 df-mpt 4045 df-id 4271 df-xp 4610 df-rel 4611 df-cnv 4612 df-co 4613 df-dm 4614 df-rn 4615 df-res 4616 df-ima 4617 df-iota 5153 df-fun 5190 df-fn 5191 df-f 5192 df-f1 5193 df-fo 5194 df-f1o 5195 df-fv 5196 df-ov 5845 df-oprab 5846 df-mpo 5847 df-rest 12558 |
This theorem is referenced by: (None) |
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