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Mirrors > Home > MPE Home > Th. List > tgidm | Structured version Visualization version 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 | fvex 6201 | . . . . 5 | |
2 | eltg3 20766 | . . . . 5 | |
3 | 1, 2 | ax-mp 5 | . . . 4 |
4 | uniiun 4573 | . . . . . . . . . 10 | |
5 | simpr 477 | . . . . . . . . . . . . 13 | |
6 | 5 | sselda 3603 | . . . . . . . . . . . 12 |
7 | eltg4i 20764 | . . . . . . . . . . . 12 | |
8 | 6, 7 | syl 17 | . . . . . . . . . . 11 |
9 | 8 | iuneq2dv 4542 | . . . . . . . . . 10 |
10 | 4, 9 | syl5eq 2668 | . . . . . . . . 9 |
11 | iuncom4 4528 | . . . . . . . . 9 | |
12 | 10, 11 | syl6eq 2672 | . . . . . . . 8 |
13 | inss1 3833 | . . . . . . . . . . . 12 | |
14 | 13 | rgenw 2924 | . . . . . . . . . . 11 |
15 | iunss 4561 | . . . . . . . . . . 11 | |
16 | 14, 15 | mpbir 221 | . . . . . . . . . 10 |
17 | 16 | a1i 11 | . . . . . . . . 9 |
18 | eltg3i 20765 | . . . . . . . . 9 | |
19 | 17, 18 | sylan2 491 | . . . . . . . 8 |
20 | 12, 19 | eqeltrd 2701 | . . . . . . 7 |
21 | eleq1 2689 | . . . . . . 7 | |
22 | 20, 21 | syl5ibrcom 237 | . . . . . 6 |
23 | 22 | expimpd 629 | . . . . 5 |
24 | 23 | exlimdv 1861 | . . . 4 |
25 | 3, 24 | syl5bi 232 | . . 3 |
26 | 25 | ssrdv 3609 | . 2 |
27 | bastg 20770 | . . 3 | |
28 | tgss 20772 | . . 3 | |
29 | 1, 27, 28 | sylancr 695 | . 2 |
30 | 26, 29 | eqssd 3620 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 wceq 1483 wex 1704 wcel 1990 wral 2912 cvv 3200 cin 3573 wss 3574 cpw 4158 cuni 4436 ciun 4520 cfv 5888 ctg 16098 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-iota 5851 df-fun 5890 df-fv 5896 df-topgen 16104 |
This theorem is referenced by: tgss3 20790 txbasval 21409 |
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