Theorem istps 15056

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 15055	. . 3 |- TopSp = { f | ( TopOpen ` f ) e. (TopOn ` ( Base ` f ) ) }
2	1	eleq2i 2305	. 2 |- ( K e. TopSp <-> K e. { f | ( TopOpen ` f ) e. (TopOn ` ( Base ` f ) ) } )
3		topontop 15038	. . . 4 |- ( J e. (TopOn ` A ) -> J e. Top )
4		topnfn 13575	. . . . . . 7 |- TopOpen Fn _V
5		fnrel 5474	. . . . . . 7 |- ( TopOpen Fn _V -> Rel TopOpen )
6	4, 5	ax-mp 5	. . . . . 6 |- Rel TopOpen
7		0opn 15030	. . . . . . 7 |- ( J e. Top -> (/) e. J )
8		istps.j	. . . . . . 7 |- J = ( TopOpen ` K )
9	7, 8	eleqtrdi 2331	. . . . . 6 |- ( J e. Top -> (/) e. ( TopOpen ` K ) )
10		relelfvdm 5722	. . . . . 6 |- ( ( Rel TopOpen /\ (/) e. ( TopOpen ` K ) ) -> K e. dom TopOpen )
11	6, 9, 10	sylancr 418	. . . . 5 |- ( J e. Top -> K e. dom TopOpen )
12	11	elexd 2835	. . . 4 |- ( J e. Top -> K e. _V )
13	3, 12	syl 14	. . 3 |- ( J e. (TopOn ` A ) -> K e. _V )
14		fveq2 5690	. . . . 5 |- ( f = K -> ( TopOpen ` f ) = ( TopOpen ` K ) )
15	14, 8	eqtr4di 2289	. . . 4 |- ( f = K -> ( TopOpen ` f ) = J )
16		fveq2 5690	. . . . . 6 |- ( f = K -> ( Base ` f ) = ( Base ` K ) )
17		istps.a	. . . . . 6 |- A = ( Base ` K )
18	16, 17	eqtr4di 2289	. . . . 5 |- ( f = K -> ( Base ` f ) = A )
19	18	fveq2d 5694	. . . 4 |- ( f = K -> (TopOn ` ( Base ` f ) ) = (TopOn ` A ) )
20	15, 19	eleq12d 2309	. . 3 |- ( f = K -> ( ( TopOpen ` f ) e. (TopOn ` ( Base ` f ) ) <-> J e. (TopOn ` A ) ) )
21	13, 20	elab3 2978	. 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 1402   e. wcel 2209   {cab 2224   _Vcvv 2821   (/)c0 3520   dom cdm 4769   Rel wrel 4774   Fn wfn 5367   ` cfv 5372   Basecbs 13330   TopOpenctopn 13571   Topctop 15021  TopOnctopon 15034   TopSpctps 15054

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-cnex 8260  ax-resscn 8261  ax-1re 8263  ax-addrcl 8266

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-inn 9284  df-2 9342  df-3 9343  df-4 9344  df-5 9345  df-6 9346  df-7 9347  df-8 9348  df-9 9349  df-ndx 13333  df-slot 13334  df-base 13336  df-tset 13427  df-rest 13572  df-topn 13573  df-top 15022  df-topon 15035  df-topsp 15055

This theorem is referenced by:  istps2  15057  tpspropd  15060  tsettps  15062  isxms2  15476  cnfldtopon  15564