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Mirrors > Home > ILE Home > Th. List > txrest | Unicode version |
Description: The subspace of a topological product space induced by a subset with a Cartesian product representation is a topological product of the subspaces induced by the subspaces of the terms of the products. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 2-Sep-2015.) |
Ref | Expression |
---|---|
txrest | ↾t ↾t ↾t |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2139 | . . . . . 6 | |
2 | 1 | txval 12424 | . . . . 5 |
3 | 2 | adantr 274 | . . . 4 |
4 | 3 | oveq1d 5789 | . . 3 ↾t ↾t |
5 | 1 | txbasex 12426 | . . . 4 |
6 | xpexg 4653 | . . . 4 | |
7 | tgrest 12338 | . . . 4 ↾t ↾t | |
8 | 5, 6, 7 | syl2an 287 | . . 3 ↾t ↾t |
9 | elrest 12127 | . . . . . . . 8 ↾t | |
10 | 5, 6, 9 | syl2an 287 | . . . . . . 7 ↾t |
11 | vex 2689 | . . . . . . . . . . 11 | |
12 | 11 | inex1 4062 | . . . . . . . . . 10 |
13 | 12 | a1i 9 | . . . . . . . . 9 |
14 | elrest 12127 | . . . . . . . . . 10 ↾t | |
15 | 14 | ad2ant2r 500 | . . . . . . . . 9 ↾t |
16 | xpeq1 4553 | . . . . . . . . . . . 12 | |
17 | 16 | eqeq2d 2151 | . . . . . . . . . . 11 |
18 | 17 | rexbidv 2438 | . . . . . . . . . 10 ↾t ↾t |
19 | vex 2689 | . . . . . . . . . . . . 13 | |
20 | 19 | inex1 4062 | . . . . . . . . . . . 12 |
21 | 20 | a1i 9 | . . . . . . . . . . 11 |
22 | elrest 12127 | . . . . . . . . . . . 12 ↾t | |
23 | 22 | ad2ant2l 499 | . . . . . . . . . . 11 ↾t |
24 | xpeq2 4554 | . . . . . . . . . . . . 13 | |
25 | 24 | eqeq2d 2151 | . . . . . . . . . . . 12 |
26 | 25 | adantl 275 | . . . . . . . . . . 11 |
27 | 21, 23, 26 | rexxfr2d 4386 | . . . . . . . . . 10 ↾t |
28 | 18, 27 | sylan9bbr 458 | . . . . . . . . 9 ↾t |
29 | 13, 15, 28 | rexxfr2d 4386 | . . . . . . . 8 ↾t ↾t |
30 | 11, 19 | xpex 4654 | . . . . . . . . . 10 |
31 | 30 | rgen2w 2488 | . . . . . . . . 9 |
32 | eqid 2139 | . . . . . . . . . 10 | |
33 | ineq1 3270 | . . . . . . . . . . . 12 | |
34 | inxp 4673 | . . . . . . . . . . . 12 | |
35 | 33, 34 | syl6eq 2188 | . . . . . . . . . . 11 |
36 | 35 | eqeq2d 2151 | . . . . . . . . . 10 |
37 | 32, 36 | rexrnmpo 5886 | . . . . . . . . 9 |
38 | 31, 37 | ax-mp 5 | . . . . . . . 8 |
39 | 29, 38 | syl6bbr 197 | . . . . . . 7 ↾t ↾t |
40 | 10, 39 | bitr4d 190 | . . . . . 6 ↾t ↾t ↾t |
41 | 40 | abbi2dv 2258 | . . . . 5 ↾t ↾t ↾t |
42 | eqid 2139 | . . . . . 6 ↾t ↾t ↾t ↾t | |
43 | 42 | rnmpo 5881 | . . . . 5 ↾t ↾t ↾t ↾t |
44 | 41, 43 | syl6eqr 2190 | . . . 4 ↾t ↾t ↾t |
45 | 44 | fveq2d 5425 | . . 3 ↾t ↾t ↾t |
46 | 4, 8, 45 | 3eqtr2d 2178 | . 2 ↾t ↾t ↾t |
47 | restfn 12124 | . . . 4 ↾t | |
48 | simpll 518 | . . . . 5 | |
49 | 48 | elexd 2699 | . . . 4 |
50 | simprl 520 | . . . . 5 | |
51 | 50 | elexd 2699 | . . . 4 |
52 | fnovex 5804 | . . . 4 ↾t ↾t | |
53 | 47, 49, 51, 52 | mp3an2i 1320 | . . 3 ↾t |
54 | simplr 519 | . . . . 5 | |
55 | 54 | elexd 2699 | . . . 4 |
56 | simprr 521 | . . . . 5 | |
57 | 56 | elexd 2699 | . . . 4 |
58 | fnovex 5804 | . . . 4 ↾t ↾t | |
59 | 47, 55, 57, 58 | mp3an2i 1320 | . . 3 ↾t |
60 | eqid 2139 | . . . 4 ↾t ↾t ↾t ↾t | |
61 | 60 | txval 12424 | . . 3 ↾t ↾t ↾t ↾t ↾t ↾t |
62 | 53, 59, 61 | syl2anc 408 | . 2 ↾t ↾t ↾t ↾t |
63 | 46, 62 | eqtr4d 2175 | 1 ↾t ↾t ↾t |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1331 wcel 1480 cab 2125 wral 2416 wrex 2417 cvv 2686 cin 3070 cxp 4537 crn 4540 wfn 5118 cfv 5123 (class class class)co 5774 cmpo 5776 ↾t crest 12120 ctg 12135 ctx 12421 |
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 df-topgen 12141 df-tx 12422 |
This theorem is referenced by: cnmpt2res 12466 limccnp2cntop 12815 |
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