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Mirrors > Home > ILE Home > Th. List > istopon | Unicode version |
Description: Property of being a topology with a given base set. (Contributed by Stefan O'Rear, 31-Jan-2015.) (Revised by Mario Carneiro, 13-Aug-2015.) |
Ref | Expression |
---|---|
istopon | TopOn |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funtopon 12179 | . . . . 5 TopOn | |
2 | funrel 5140 | . . . . 5 TopOn TopOn | |
3 | 1, 2 | ax-mp 5 | . . . 4 TopOn |
4 | relelfvdm 5453 | . . . 4 TopOn TopOn TopOn | |
5 | 3, 4 | mpan 420 | . . 3 TopOn TopOn |
6 | 5 | elexd 2699 | . 2 TopOn |
7 | uniexg 4361 | . . . 4 | |
8 | eleq1 2202 | . . . 4 | |
9 | 7, 8 | syl5ibrcom 156 | . . 3 |
10 | 9 | imp 123 | . 2 |
11 | eqeq1 2146 | . . . . . 6 | |
12 | 11 | rabbidv 2675 | . . . . 5 |
13 | df-topon 12178 | . . . . 5 TopOn | |
14 | vpwex 4103 | . . . . . . 7 | |
15 | 14 | pwex 4107 | . . . . . 6 |
16 | rabss 3174 | . . . . . . 7 | |
17 | pwuni 4116 | . . . . . . . . . 10 | |
18 | pweq 3513 | . . . . . . . . . 10 | |
19 | 17, 18 | sseqtrrid 3148 | . . . . . . . . 9 |
20 | velpw 3517 | . . . . . . . . 9 | |
21 | 19, 20 | sylibr 133 | . . . . . . . 8 |
22 | 21 | a1i 9 | . . . . . . 7 |
23 | 16, 22 | mprgbir 2490 | . . . . . 6 |
24 | 15, 23 | ssexi 4066 | . . . . 5 |
25 | 12, 13, 24 | fvmpt3i 5501 | . . . 4 TopOn |
26 | 25 | eleq2d 2209 | . . 3 TopOn |
27 | unieq 3745 | . . . . 5 | |
28 | 27 | eqeq2d 2151 | . . . 4 |
29 | 28 | elrab 2840 | . . 3 |
30 | 26, 29 | syl6bb 195 | . 2 TopOn |
31 | 6, 10, 30 | pm5.21nii 693 | 1 TopOn |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1331 wcel 1480 crab 2420 cvv 2686 wss 3071 cpw 3510 cuni 3736 cdm 4539 wrel 4544 wfun 5117 cfv 5123 ctop 12164 TopOnctopon 12177 |
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-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-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 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-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-sbc 2910 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-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-iota 5088 df-fun 5125 df-fv 5131 df-topon 12178 |
This theorem is referenced by: topontop 12181 toponuni 12182 toptopon 12185 toponcom 12194 istps2 12200 tgtopon 12235 distopon 12256 epttop 12259 resttopon 12340 resttopon2 12347 txtopon 12431 |
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