Theorem istps 12036

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 12035	. . 3 |- TopSp = { f | ( TopOpen ` f ) e. (TopOn ` ( Base ` f ) ) }
2	1	eleq2i 2179	. 2 |- ( K e. TopSp <-> K e. { f | ( TopOpen ` f ) e. (TopOn ` ( Base ` f ) ) } )
3		topontop 12018	. . . 4 |- ( J e. (TopOn ` A ) -> J e. Top )
4		topnfn 11962	. . . . . . 7 |- TopOpen Fn _V
5		fnrel 5177	. . . . . . 7 |- ( TopOpen Fn _V -> Rel TopOpen )
6	4, 5	ax-mp 7	. . . . . 6 |- Rel TopOpen
7		0opn 12010	. . . . . . 7 |- ( J e. Top -> (/) e. J )
8		istps.j	. . . . . . 7 |- J = ( TopOpen ` K )
9	7, 8	syl6eleq 2205	. . . . . 6 |- ( J e. Top -> (/) e. ( TopOpen ` K ) )
10		relelfvdm 5405	. . . . . 6 |- ( ( Rel TopOpen /\ (/) e. ( TopOpen ` K ) ) -> K e. dom TopOpen )
11	6, 9, 10	sylancr 408	. . . . 5 |- ( J e. Top -> K e. dom TopOpen )
12	11	elexd 2668	. . . 4 |- ( J e. Top -> K e. _V )
13	3, 12	syl 14	. . 3 |- ( J e. (TopOn ` A ) -> K e. _V )
14		fveq2 5373	. . . . 5 |- ( f = K -> ( TopOpen ` f ) = ( TopOpen ` K ) )
15	14, 8	syl6eqr 2163	. . . 4 |- ( f = K -> ( TopOpen ` f ) = J )
16		fveq2 5373	. . . . . 6 |- ( f = K -> ( Base ` f ) = ( Base ` K ) )
17		istps.a	. . . . . 6 |- A = ( Base ` K )
18	16, 17	syl6eqr 2163	. . . . 5 |- ( f = K -> ( Base ` f ) = A )
19	18	fveq2d 5377	. . . 4 |- ( f = K -> (TopOn ` ( Base ` f ) ) = (TopOn ` A ) )
20	15, 19	eleq12d 2183	. . 3 |- ( f = K -> ( ( TopOpen ` f ) e. (TopOn ` ( Base ` f ) ) <-> J e. (TopOn ` A ) ) )
21	13, 20	elab3 2803	. 2 |- ( K e. { f | ( TopOpen ` f ) e. (TopOn ` ( Base ` f ) ) } <-> J e. (TopOn ` A ) )
22	2, 21	bitri 183	1 |- ( K e. TopSp <-> J e. (TopOn ` A ) )

Syntax hints:   <-> wb 104   = wceq 1312   e. wcel 1461   {cab 2099   _Vcvv 2655   (/)c0 3327   dom cdm 4497   Rel wrel 4502   Fn wfn 5074   ` cfv 5079   Basecbs 11796   TopOpenctopn 11958   Topctop 12001  TopOnctopon 12014   TopSpctps 12034

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-13 1472  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-coll 4001  ax-sep 4004  ax-pow 4056  ax-pr 4089  ax-un 4313  ax-cnex 7630  ax-resscn 7631  ax-1re 7633  ax-addrcl 7636

This theorem depends on definitions:  df-bi 116  df-3an 945  df-tru 1315  df-fal 1318  df-nf 1418  df-sb 1717  df-eu 1976  df-mo 1977  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ral 2393  df-rex 2394  df-reu 2395  df-rab 2397  df-v 2657  df-sbc 2877  df-csb 2970  df-dif 3037  df-un 3039  df-in 3041  df-ss 3048  df-nul 3328  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-uni 3701  df-int 3736  df-iun 3779  df-br 3894  df-opab 3948  df-mpt 3949  df-id 4173  df-xp 4503  df-rel 4504  df-cnv 4505  df-co 4506  df-dm 4507  df-rn 4508  df-res 4509  df-ima 4510  df-iota 5044  df-fun 5081  df-fn 5082  df-f 5083  df-f1 5084  df-fo 5085  df-f1o 5086  df-fv 5087  df-ov 5729  df-oprab 5730  df-mpo 5731  df-1st 5990  df-2nd 5991  df-inn 8625  df-2 8683  df-3 8684  df-4 8685  df-5 8686  df-6 8687  df-7 8688  df-8 8689  df-9 8690  df-ndx 11799  df-slot 11800  df-base 11802  df-tset 11877  df-rest 11959  df-topn 11960  df-top 12002  df-topon 12015  df-topsp 12035

This theorem is referenced by:  istps2  12037  tpspropd  12040  tsettps  12042  isxms2  12435