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Mirrors > Home > ILE Home > Th. List > tgidm | Unicode version |
Description: The topology generator function is idempotent. (Contributed by NM, 18-Jul-2006.) (Revised by Mario Carneiro, 2-Sep-2015.) |
Ref | Expression |
---|---|
tgidm |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tgvalex 12222 | . . . . 5 | |
2 | eltg3 12229 | . . . . 5 | |
3 | 1, 2 | syl 14 | . . . 4 |
4 | uniiun 3866 | . . . . . . . . . 10 | |
5 | simpr 109 | . . . . . . . . . . . . 13 | |
6 | 5 | sselda 3097 | . . . . . . . . . . . 12 |
7 | eltg4i 12227 | . . . . . . . . . . . 12 | |
8 | 6, 7 | syl 14 | . . . . . . . . . . 11 |
9 | 8 | iuneq2dv 3834 | . . . . . . . . . 10 |
10 | 4, 9 | syl5eq 2184 | . . . . . . . . 9 |
11 | iuncom4 3820 | . . . . . . . . 9 | |
12 | 10, 11 | syl6eq 2188 | . . . . . . . 8 |
13 | inss1 3296 | . . . . . . . . . . . 12 | |
14 | 13 | rgenw 2487 | . . . . . . . . . . 11 |
15 | iunss 3854 | . . . . . . . . . . 11 | |
16 | 14, 15 | mpbir 145 | . . . . . . . . . 10 |
17 | 16 | a1i 9 | . . . . . . . . 9 |
18 | eltg3i 12228 | . . . . . . . . 9 | |
19 | 17, 18 | sylan2 284 | . . . . . . . 8 |
20 | 12, 19 | eqeltrd 2216 | . . . . . . 7 |
21 | eleq1 2202 | . . . . . . 7 | |
22 | 20, 21 | syl5ibrcom 156 | . . . . . 6 |
23 | 22 | expimpd 360 | . . . . 5 |
24 | 23 | exlimdv 1791 | . . . 4 |
25 | 3, 24 | sylbid 149 | . . 3 |
26 | 25 | ssrdv 3103 | . 2 |
27 | bastg 12233 | . . 3 | |
28 | tgss 12235 | . . 3 | |
29 | 1, 27, 28 | syl2anc 408 | . 2 |
30 | 26, 29 | eqssd 3114 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1331 wex 1468 wcel 1480 wral 2416 cvv 2686 cin 3070 wss 3071 cpw 3510 cuni 3736 ciun 3813 cfv 5123 ctg 12138 |
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-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-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-iota 5088 df-fun 5125 df-fv 5131 df-topgen 12144 |
This theorem is referenced by: tgss3 12250 txbasval 12439 |
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