Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > restin | Unicode version |
Description: When the subspace region is not a subset of the base of the topology, the resulting set is the same as the subspace restricted to the base. (Contributed by Mario Carneiro, 15-Dec-2013.) |
Ref | Expression |
---|---|
restin.1 |
Ref | Expression |
---|---|
restin | ↾t ↾t |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | restin.1 | . . . . 5 | |
2 | uniexg 4361 | . . . . 5 | |
3 | 1, 2 | eqeltrid 2226 | . . . 4 |
4 | 3 | adantr 274 | . . 3 |
5 | restco 12343 | . . . 4 ↾t ↾t ↾t | |
6 | 5 | 3com23 1187 | . . 3 ↾t ↾t ↾t |
7 | 4, 6 | mpd3an3 1316 | . 2 ↾t ↾t ↾t |
8 | 1 | restid 12131 | . . . 4 ↾t |
9 | 8 | adantr 274 | . . 3 ↾t |
10 | 9 | oveq1d 5789 | . 2 ↾t ↾t ↾t |
11 | incom 3268 | . . . 4 | |
12 | 11 | oveq2i 5785 | . . 3 ↾t ↾t |
13 | 12 | a1i 9 | . 2 ↾t ↾t |
14 | 7, 10, 13 | 3eqtr3d 2180 | 1 ↾t ↾t |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1331 wcel 1480 cvv 2686 cin 3070 cuni 3736 (class class class)co 5774 ↾t crest 12120 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-rest 12122 |
This theorem is referenced by: restuni2 12346 cnrest2r 12406 |
Copyright terms: Public domain | W3C validator |