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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 12999 | . . 3 |
3 | pwexg 4175 | . . 3 | |
4 | mptexg 5733 | . . 3 | |
5 | 2, 3, 4 | 3syl 17 | . 2 |
6 | unieq 3814 | . . . . . 6 | |
7 | 6, 1 | eqtr4di 2226 | . . . . 5 |
8 | 7 | pweqd 3577 | . . . 4 |
9 | fveq2 5507 | . . . . . 6 | |
10 | rabeq 2727 | . . . . . 6 | |
11 | 9, 10 | syl 14 | . . . . 5 |
12 | 11 | inteqd 3845 | . . . 4 |
13 | 8, 12 | mpteq12dv 4080 | . . 3 |
14 | df-cls 13090 | . . 3 | |
15 | 13, 14 | fvmptg 5584 | . 2 |
16 | 5, 15 | mpdan 421 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wceq 1353 wcel 2146 crab 2457 cvv 2735 wss 3127 cpw 3572 cuni 3805 cint 3840 cmpt 4059 cfv 5208 ctop 12988 ccld 13085 ccl 13087 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1445 ax-7 1446 ax-gen 1447 ax-ie1 1491 ax-ie2 1492 ax-8 1502 ax-10 1503 ax-11 1504 ax-i12 1505 ax-bndl 1507 ax-4 1508 ax-17 1524 ax-i9 1528 ax-ial 1532 ax-i5r 1533 ax-14 2149 ax-ext 2157 ax-coll 4113 ax-sep 4116 ax-pow 4169 ax-pr 4203 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1459 df-sb 1761 df-eu 2027 df-mo 2028 df-clab 2162 df-cleq 2168 df-clel 2171 df-nfc 2306 df-ral 2458 df-rex 2459 df-reu 2460 df-rab 2462 df-v 2737 df-sbc 2961 df-csb 3056 df-un 3131 df-in 3133 df-ss 3140 df-pw 3574 df-sn 3595 df-pr 3596 df-op 3598 df-uni 3806 df-int 3841 df-iun 3884 df-br 3999 df-opab 4060 df-mpt 4061 df-id 4287 df-xp 4626 df-rel 4627 df-cnv 4628 df-co 4629 df-dm 4630 df-rn 4631 df-res 4632 df-ima 4633 df-iota 5170 df-fun 5210 df-fn 5211 df-f 5212 df-f1 5213 df-fo 5214 df-f1o 5215 df-fv 5216 df-top 12989 df-cls 13090 |
This theorem is referenced by: clsval 13104 |
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