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Mirrors > Home > MPE Home > Th. List > istopon | Structured version Visualization version 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 | elfvex 6221 | . 2 TopOn | |
2 | uniexg 6955 | . . . 4 | |
3 | eleq1 2689 | . . . 4 | |
4 | 2, 3 | syl5ibrcom 237 | . . 3 |
5 | 4 | imp 445 | . 2 |
6 | eqeq1 2626 | . . . . . 6 | |
7 | 6 | rabbidv 3189 | . . . . 5 |
8 | df-topon 20716 | . . . . 5 TopOn | |
9 | vpwex 4849 | . . . . . . 7 | |
10 | 9 | pwex 4848 | . . . . . 6 |
11 | rabss 3679 | . . . . . . 7 | |
12 | pwuni 4474 | . . . . . . . . . 10 | |
13 | pweq 4161 | . . . . . . . . . 10 | |
14 | 12, 13 | syl5sseqr 3654 | . . . . . . . . 9 |
15 | selpw 4165 | . . . . . . . . 9 | |
16 | 14, 15 | sylibr 224 | . . . . . . . 8 |
17 | 16 | a1i 11 | . . . . . . 7 |
18 | 11, 17 | mprgbir 2927 | . . . . . 6 |
19 | 10, 18 | ssexi 4803 | . . . . 5 |
20 | 7, 8, 19 | fvmpt3i 6287 | . . . 4 TopOn |
21 | 20 | eleq2d 2687 | . . 3 TopOn |
22 | unieq 4444 | . . . . 5 | |
23 | 22 | eqeq2d 2632 | . . . 4 |
24 | 23 | elrab 3363 | . . 3 |
25 | 21, 24 | syl6bb 276 | . 2 TopOn |
26 | 1, 5, 25 | pm5.21nii 368 | 1 TopOn |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 wceq 1483 wcel 1990 crab 2916 cvv 3200 wss 3574 cpw 4158 cuni 4436 cfv 5888 ctop 20698 TopOnctopon 20715 |
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-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-topon 20716 |
This theorem is referenced by: topontop 20718 toponuni 20719 toptopon 20722 toponcom 20732 istps2 20739 tgtopon 20775 distopon 20801 indistopon 20805 fctop 20808 cctop 20810 ppttop 20811 epttop 20813 mretopd 20896 toponmre 20897 resttopon 20965 resttopon2 20972 kgentopon 21341 txtopon 21394 pttopon 21399 xkotopon 21403 qtoptopon 21507 flimtopon 21774 fclstopon 21816 fclsfnflim 21831 utoptopon 22040 qtopt1 29902 neibastop1 32354 onsuctopon 32433 rfcnpre1 39178 cnfex 39187 icccncfext 40100 stoweidlem47 40264 |
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