Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > clsfval | Unicode version |
Description: The closure function on the subsets of a topology's base set. (Contributed by NM, 3-Oct-2006.) (Revised by Mario Carneiro, 11-Nov-2013.) |
Ref | Expression |
---|---|
cldval.1 |
Ref | Expression |
---|---|
clsfval |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cldval.1 | . . . 4 | |
2 | 1 | topopn 12553 | . . 3 |
3 | pwexg 4153 | . . 3 | |
4 | mptexg 5704 | . . 3 | |
5 | 2, 3, 4 | 3syl 17 | . 2 |
6 | unieq 3792 | . . . . . 6 | |
7 | 6, 1 | eqtr4di 2215 | . . . . 5 |
8 | 7 | pweqd 3558 | . . . 4 |
9 | fveq2 5480 | . . . . . 6 | |
10 | rabeq 2713 | . . . . . 6 | |
11 | 9, 10 | syl 14 | . . . . 5 |
12 | 11 | inteqd 3823 | . . . 4 |
13 | 8, 12 | mpteq12dv 4058 | . . 3 |
14 | df-cls 12644 | . . 3 | |
15 | 13, 14 | fvmptg 5556 | . 2 |
16 | 5, 15 | mpdan 418 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wceq 1342 wcel 2135 crab 2446 cvv 2721 wss 3111 cpw 3553 cuni 3783 cint 3818 cmpt 4037 cfv 5182 ctop 12542 ccld 12639 ccl 12641 |
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 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-coll 4091 ax-sep 4094 ax-pow 4147 ax-pr 4181 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-top 12543 df-cls 12644 |
This theorem is referenced by: clsval 12658 |
Copyright terms: Public domain | W3C validator |