Theorem istopon 11880

Description: Property of being a topology with a given base set. (Contributed by Stefan O'Rear, 31-Jan-2015.) (Revised by Mario Carneiro, 13-Aug-2015.)

Assertion

Ref	Expression
istopon	|- ( J e. (TopOn ` B ) <-> ( J e. Top /\ B = U. J ) )

Proof of Theorem istopon

Dummy variables b j are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		funtopon 11879	. . . . 5 |- Fun TopOn
2		funrel 5066	. . . . 5 |- ( Fun TopOn -> Rel TopOn )
3	1, 2	ax-mp 7	. . . 4 |- Rel TopOn
4		relelfvdm 5371	. . . 4 |- ( ( Rel TopOn /\ J e. (TopOn ` B ) ) -> B e. dom TopOn )
5	3, 4	mpan 416	. . 3 |- ( J e. (TopOn ` B ) -> B e. dom TopOn )
6	5	elexd 2646	. 2 |- ( J e. (TopOn ` B ) -> B e. _V )
7		uniexg 4290	. . . 4 |- ( J e. Top -> U. J e. _V )
8		eleq1 2157	. . . 4 |- ( B = U. J -> ( B e. _V <-> U. J e. _V ) )
9	7, 8	syl5ibrcom 156	. . 3 |- ( J e. Top -> ( B = U. J -> B e. _V ) )
10	9	imp 123	. 2 |- ( ( J e. Top /\ B = U. J ) -> B e. _V )
11		eqeq1 2101	. . . . . 6 |- ( b = B -> ( b = U. j <-> B = U. j ) )
12	11	rabbidv 2622	. . . . 5 |- ( b = B -> { j e. Top | b = U. j } = { j e. Top | B = U. j } )
13		df-topon 11878	. . . . 5 |- TopOn = ( b e. _V |-> { j e. Top | b = U. j } )
14		vpwex 4035	. . . . . . 7 |- ~P b e. _V
15	14	pwex 4039	. . . . . 6 |- ~P ~P b e. _V
16		rabss 3113	. . . . . . 7 |- ( { j e. Top | b = U. j } C_ ~P ~P b <-> A. j e. Top ( b = U. j -> j e. ~P ~P b ) )
17		pwuni 4048	. . . . . . . . . 10 |- j C_ ~P U. j
18		pweq 3452	. . . . . . . . . 10 |- ( b = U. j -> ~P b = ~P U. j )
19	17, 18	syl5sseqr 3090	. . . . . . . . 9 |- ( b = U. j -> j C_ ~P b )
20		selpw 3456	. . . . . . . . 9 |- ( j e. ~P ~P b <-> j C_ ~P b )
21	19, 20	sylibr 133	. . . . . . . 8 |- ( b = U. j -> j e. ~P ~P b )
22	21	a1i 9	. . . . . . 7 |- ( j e. Top -> ( b = U. j -> j e. ~P ~P b ) )
23	16, 22	mprgbir 2444	. . . . . 6 |- { j e. Top | b = U. j } C_ ~P ~P b
24	15, 23	ssexi 3998	. . . . 5 |- { j e. Top | b = U. j } e. _V
25	12, 13, 24	fvmpt3i 5419	. . . 4 |- ( B e. _V -> (TopOn ` B ) = { j e. Top | B = U. j } )
26	25	eleq2d 2164	. . 3 |- ( B e. _V -> ( J e. (TopOn ` B ) <-> J e. { j e. Top | B = U. j } ) )
27		unieq 3684	. . . . 5 |- ( j = J -> U. j = U. J )
28	27	eqeq2d 2106	. . . 4 |- ( j = J -> ( B = U. j <-> B = U. J ) )
29	28	elrab 2785	. . 3 |- ( J e. { j e. Top | B = U. j } <-> ( J e. Top /\ B = U. J ) )
30	26, 29	syl6bb 195	. 2 |- ( B e. _V -> ( J e. (TopOn ` B ) <-> ( J e. Top /\ B = U. J ) ) )
31	6, 10, 30	pm5.21nii 658	1 |- ( J e. (TopOn ` B ) <-> ( J e. Top /\ B = U. J ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1296   e. wcel 1445   {crab 2374   _Vcvv 2633   C_ wss 3013   ~Pcpw 3449   U.cuni 3675   dom cdm 4467   Rel wrel 4472   Fun wfun 5043   ` cfv 5049   Topctop 11864  TopOnctopon 11877

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-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-pow 4030  ax-pr 4060  ax-un 4284

This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-rex 2376  df-rab 2379  df-v 2635  df-sbc 2855  df-un 3017  df-in 3019  df-ss 3026  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-br 3868  df-opab 3922  df-mpt 3923  df-id 4144  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-iota 5014  df-fun 5051  df-fv 5057  df-topon 11878

This theorem is referenced by:  topontop  11881  toponuni  11882  toptopon  11885  toponcom  11893  istps2  11899  tgtopon  11934  distopon  11955  epttop  11958  resttopon  12039  resttopon2  12046