Theorem istopon 15037

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 15036	. . . . 5 |- Fun TopOn
2		funrel 5389	. . . . 5 |- ( Fun TopOn -> Rel TopOn )
3	1, 2	ax-mp 5	. . . 4 |- Rel TopOn
4		relelfvdm 5722	. . . 4 |- ( ( Rel TopOn /\ J e. (TopOn ` B ) ) -> B e. dom TopOn )
5	3, 4	mpan 428	. . 3 |- ( J e. (TopOn ` B ) -> B e. dom TopOn )
6	5	elexd 2835	. 2 |- ( J e. (TopOn ` B ) -> B e. _V )
7		uniexg 4580	. . . 4 |- ( J e. Top -> U. J e. _V )
8		eleq1 2301	. . . 4 |- ( B = U. J -> ( B e. _V <-> U. J e. _V ) )
9	7, 8	syl5ibrcom 157	. . 3 |- ( J e. Top -> ( B = U. J -> B e. _V ) )
10	9	imp 124	. 2 |- ( ( J e. Top /\ B = U. J ) -> B e. _V )
11		eqeq1 2245	. . . . . 6 |- ( b = B -> ( b = U. j <-> B = U. j ) )
12	11	rabbidv 2810	. . . . 5 |- ( b = B -> { j e. Top | b = U. j } = { j e. Top | B = U. j } )
13		df-topon 15035	. . . . 5 |- TopOn = ( b e. _V |-> { j e. Top | b = U. j } )
14		vpwex 4311	. . . . . . 7 |- ~P b e. _V
15	14	pwex 4315	. . . . . 6 |- ~P ~P b e. _V
16		rabss 3325	. . . . . . 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 4324	. . . . . . . . . 10 |- j C_ ~P U. j
18		pweq 3688	. . . . . . . . . 10 |- ( b = U. j -> ~P b = ~P U. j )
19	17, 18	sseqtrrid 3299	. . . . . . . . 9 |- ( b = U. j -> j C_ ~P b )
20		velpw 3692	. . . . . . . . 9 |- ( j e. ~P ~P b <-> j C_ ~P b )
21	19, 20	sylibr 134	. . . . . . . 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 2608	. . . . . 6 |- { j e. Top | b = U. j } C_ ~P ~P b
24	15, 23	ssexi 4266	. . . . 5 |- { j e. Top | b = U. j } e. _V
25	12, 13, 24	fvmpt3i 5779	. . . 4 |- ( B e. _V -> (TopOn ` B ) = { j e. Top | B = U. j } )
26	25	eleq2d 2308	. . 3 |- ( B e. _V -> ( J e. (TopOn ` B ) <-> J e. { j e. Top | B = U. j } ) )
27		unieq 3939	. . . . 5 |- ( j = J -> U. j = U. J )
28	27	eqeq2d 2250	. . . 4 |- ( j = J -> ( B = U. j <-> B = U. J ) )
29	28	elrab 2982	. . 3 |- ( J e. { j e. Top | B = U. j } <-> ( J e. Top /\ B = U. J ) )
30	26, 29	bitrdi 196	. 2 |- ( B e. _V -> ( J e. (TopOn ` B ) <-> ( J e. Top /\ B = U. J ) ) )
31	6, 10, 30	pm5.21nii 716	1 |- ( J e. (TopOn ` B ) <-> ( J e. Top /\ B = U. J ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   {crab 2532   _Vcvv 2821   C_ wss 3220   ~Pcpw 3685   U.cuni 3930   dom cdm 4769   Rel wrel 4774   Fun wfun 5366   ` cfv 5372   Topctop 15021  TopOnctopon 15034

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-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-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  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-rab 2537  df-v 2823  df-sbc 3052  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  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-iota 5332  df-fun 5374  df-fv 5380  df-topon 15035

This theorem is referenced by:  topontop  15038  toponuni  15039  toptopon  15042  toponcom  15051  istps2  15057  tgtopon  15090  distopon  15111  epttop  15114  resttopon  15195  resttopon2  15202  txtopon  15286