Theorem istopon 12651

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 12650	. . . . 5 |- Fun TopOn
2		funrel 5205	. . . . 5 |- ( Fun TopOn -> Rel TopOn )
3	1, 2	ax-mp 5	. . . 4 |- Rel TopOn
4		relelfvdm 5518	. . . 4 |- ( ( Rel TopOn /\ J e. (TopOn ` B ) ) -> B e. dom TopOn )
5	3, 4	mpan 421	. . 3 |- ( J e. (TopOn ` B ) -> B e. dom TopOn )
6	5	elexd 2739	. 2 |- ( J e. (TopOn ` B ) -> B e. _V )
7		uniexg 4417	. . . 4 |- ( J e. Top -> U. J e. _V )
8		eleq1 2229	. . . 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 2172	. . . . . 6 |- ( b = B -> ( b = U. j <-> B = U. j ) )
12	11	rabbidv 2715	. . . . 5 |- ( b = B -> { j e. Top | b = U. j } = { j e. Top | B = U. j } )
13		df-topon 12649	. . . . 5 |- TopOn = ( b e. _V |-> { j e. Top | b = U. j } )
14		vpwex 4158	. . . . . . 7 |- ~P b e. _V
15	14	pwex 4162	. . . . . 6 |- ~P ~P b e. _V
16		rabss 3219	. . . . . . 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 4171	. . . . . . . . . 10 |- j C_ ~P U. j
18		pweq 3562	. . . . . . . . . 10 |- ( b = U. j -> ~P b = ~P U. j )
19	17, 18	sseqtrrid 3193	. . . . . . . . 9 |- ( b = U. j -> j C_ ~P b )
20		velpw 3566	. . . . . . . . 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 2524	. . . . . 6 |- { j e. Top | b = U. j } C_ ~P ~P b
24	15, 23	ssexi 4120	. . . . 5 |- { j e. Top | b = U. j } e. _V
25	12, 13, 24	fvmpt3i 5566	. . . 4 |- ( B e. _V -> (TopOn ` B ) = { j e. Top | B = U. j } )
26	25	eleq2d 2236	. . 3 |- ( B e. _V -> ( J e. (TopOn ` B ) <-> J e. { j e. Top | B = U. j } ) )
27		unieq 3798	. . . . 5 |- ( j = J -> U. j = U. J )
28	27	eqeq2d 2177	. . . 4 |- ( j = J -> ( B = U. j <-> B = U. J ) )
29	28	elrab 2882	. . 3 |- ( J e. { j e. Top | B = U. j } <-> ( J e. Top /\ B = U. J ) )
30	26, 29	bitrdi 195	. 2 |- ( B e. _V -> ( J e. (TopOn ` B ) <-> ( J e. Top /\ B = U. J ) ) )
31	6, 10, 30	pm5.21nii 694	1 |- ( J e. (TopOn ` B ) <-> ( J e. Top /\ B = U. J ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1343   e. wcel 2136   {crab 2448   _Vcvv 2726   C_ wss 3116   ~Pcpw 3559   U.cuni 3789   dom cdm 4604   Rel wrel 4609   Fun wfun 5182   ` cfv 5188   Topctop 12635  TopOnctopon 12648

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-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-rab 2453  df-v 2728  df-sbc 2952  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-iota 5153  df-fun 5190  df-fv 5196  df-topon 12649

This theorem is referenced by:  topontop  12652  toponuni  12653  toptopon  12656  toponcom  12665  istps2  12671  tgtopon  12706  distopon  12727  epttop  12730  resttopon  12811  resttopon2  12818  txtopon  12902