Theorem istps 14177

Description: Express the predicate "is a topological space". (Contributed by Mario Carneiro, 13-Aug-2015.)

Hypotheses

Ref	Expression
istps.a	|- A = ( Base ` K )
istps.j	|- J = ( TopOpen ` K )

Assertion

Ref	Expression
istps	|- ( K e. TopSp <-> J e. (TopOn ` A ) )

Proof of Theorem istps

Dummy variable f is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-topsp 14176	. . 3 |- TopSp = { f | ( TopOpen ` f ) e. (TopOn ` ( Base ` f ) ) }
2	1	eleq2i 2260	. 2 |- ( K e. TopSp <-> K e. { f | ( TopOpen ` f ) e. (TopOn ` ( Base ` f ) ) } )
3		topontop 14159	. . . 4 |- ( J e. (TopOn ` A ) -> J e. Top )
4		topnfn 12845	. . . . . . 7 |- TopOpen Fn _V
5		fnrel 5344	. . . . . . 7 |- ( TopOpen Fn _V -> Rel TopOpen )
6	4, 5	ax-mp 5	. . . . . 6 |- Rel TopOpen
7		0opn 14151	. . . . . . 7 |- ( J e. Top -> (/) e. J )
8		istps.j	. . . . . . 7 |- J = ( TopOpen ` K )
9	7, 8	eleqtrdi 2286	. . . . . 6 |- ( J e. Top -> (/) e. ( TopOpen ` K ) )
10		relelfvdm 5578	. . . . . 6 |- ( ( Rel TopOpen /\ (/) e. ( TopOpen ` K ) ) -> K e. dom TopOpen )
11	6, 9, 10	sylancr 414	. . . . 5 |- ( J e. Top -> K e. dom TopOpen )
12	11	elexd 2773	. . . 4 |- ( J e. Top -> K e. _V )
13	3, 12	syl 14	. . 3 |- ( J e. (TopOn ` A ) -> K e. _V )
14		fveq2 5546	. . . . 5 |- ( f = K -> ( TopOpen ` f ) = ( TopOpen ` K ) )
15	14, 8	eqtr4di 2244	. . . 4 |- ( f = K -> ( TopOpen ` f ) = J )
16		fveq2 5546	. . . . . 6 |- ( f = K -> ( Base ` f ) = ( Base ` K ) )
17		istps.a	. . . . . 6 |- A = ( Base ` K )
18	16, 17	eqtr4di 2244	. . . . 5 |- ( f = K -> ( Base ` f ) = A )
19	18	fveq2d 5550	. . . 4 |- ( f = K -> (TopOn ` ( Base ` f ) ) = (TopOn ` A ) )
20	15, 19	eleq12d 2264	. . 3 |- ( f = K -> ( ( TopOpen ` f ) e. (TopOn ` ( Base ` f ) ) <-> J e. (TopOn ` A ) ) )
21	13, 20	elab3 2912	. 2 |- ( K e. { f | ( TopOpen ` f ) e. (TopOn ` ( Base ` f ) ) } <-> J e. (TopOn ` A ) )
22	2, 21	bitri 184	1 |- ( K e. TopSp <-> J e. (TopOn ` A ) )

Syntax hints:   <-> wb 105   = wceq 1364   e. wcel 2164   {cab 2179   _Vcvv 2760   (/)c0 3446   dom cdm 4655   Rel wrel 4660   Fn wfn 5241   ` cfv 5246   Basecbs 12608   TopOpenctopn 12841   Topctop 14142  TopOnctopon 14155   TopSpctps 14175

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-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4462  ax-cnex 7953  ax-resscn 7954  ax-1re 7956  ax-addrcl 7959

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4322  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-iota 5207  df-fun 5248  df-fn 5249  df-f 5250  df-f1 5251  df-fo 5252  df-f1o 5253  df-fv 5254  df-ov 5913  df-oprab 5914  df-mpo 5915  df-1st 6184  df-2nd 6185  df-inn 8973  df-2 9031  df-3 9032  df-4 9033  df-5 9034  df-6 9035  df-7 9036  df-8 9037  df-9 9038  df-ndx 12611  df-slot 12612  df-base 12614  df-tset 12704  df-rest 12842  df-topn 12843  df-top 14143  df-topon 14156  df-topsp 14176

This theorem is referenced by:  istps2  14178  tpspropd  14181  tsettps  14183  isxms2  14597