Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > istps | Unicode version |
Description: Express the predicate "is a topological space." (Contributed by Mario Carneiro, 13-Aug-2015.) |
Ref | Expression |
---|---|
istps.a | |
istps.j |
Ref | Expression |
---|---|
istps | TopOn |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-topsp 12576 | . . 3 TopOn | |
2 | 1 | eleq2i 2231 | . 2 TopOn |
3 | topontop 12559 | . . . 4 TopOn | |
4 | topnfn 12503 | . . . . . . 7 | |
5 | fnrel 5280 | . . . . . . 7 | |
6 | 4, 5 | ax-mp 5 | . . . . . 6 |
7 | 0opn 12551 | . . . . . . 7 | |
8 | istps.j | . . . . . . 7 | |
9 | 7, 8 | eleqtrdi 2257 | . . . . . 6 |
10 | relelfvdm 5512 | . . . . . 6 | |
11 | 6, 9, 10 | sylancr 411 | . . . . 5 |
12 | 11 | elexd 2734 | . . . 4 |
13 | 3, 12 | syl 14 | . . 3 TopOn |
14 | fveq2 5480 | . . . . 5 | |
15 | 14, 8 | eqtr4di 2215 | . . . 4 |
16 | fveq2 5480 | . . . . . 6 | |
17 | istps.a | . . . . . 6 | |
18 | 16, 17 | eqtr4di 2215 | . . . . 5 |
19 | 18 | fveq2d 5484 | . . . 4 TopOn TopOn |
20 | 15, 19 | eleq12d 2235 | . . 3 TopOn TopOn |
21 | 13, 20 | elab3 2873 | . 2 TopOn TopOn |
22 | 2, 21 | bitri 183 | 1 TopOn |
Colors of variables: wff set class |
Syntax hints: wb 104 wceq 1342 wcel 2135 cab 2150 cvv 2721 c0 3404 cdm 4598 wrel 4603 wfn 5177 cfv 5182 cbs 12337 ctopn 12499 ctop 12542 TopOnctopon 12555 ctps 12575 |
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-in1 604 ax-in2 605 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-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4091 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-cnex 7835 ax-resscn 7836 ax-1re 7838 ax-addrcl 7841 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-fal 1348 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-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 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-ov 5839 df-oprab 5840 df-mpo 5841 df-1st 6100 df-2nd 6101 df-inn 8849 df-2 8907 df-3 8908 df-4 8909 df-5 8910 df-6 8911 df-7 8912 df-8 8913 df-9 8914 df-ndx 12340 df-slot 12341 df-base 12343 df-tset 12418 df-rest 12500 df-topn 12501 df-top 12543 df-topon 12556 df-topsp 12576 |
This theorem is referenced by: istps2 12578 tpspropd 12581 tsettps 12583 isxms2 12999 |
Copyright terms: Public domain | W3C validator |