Theorem istopon 12552

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 12551	. . . . 5 |- Fun TopOn
2		funrel 5199	. . . . 5 |- ( Fun TopOn -> Rel TopOn )
3	1, 2	ax-mp 5	. . . 4 |- Rel TopOn
4		relelfvdm 5512	. . . 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 2734	. 2 |- ( J e. (TopOn ` B ) -> B e. _V )
7		uniexg 4411	. . . 4 |- ( J e. Top -> U. J e. _V )
8		eleq1 2227	. . . 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 2171	. . . . . 6 |- ( b = B -> ( b = U. j <-> B = U. j ) )
12	11	rabbidv 2710	. . . . 5 |- ( b = B -> { j e. Top | b = U. j } = { j e. Top | B = U. j } )
13		df-topon 12550	. . . . 5 |- TopOn = ( b e. _V |-> { j e. Top | b = U. j } )
14		vpwex 4152	. . . . . . 7 |- ~P b e. _V
15	14	pwex 4156	. . . . . 6 |- ~P ~P b e. _V
16		rabss 3214	. . . . . . 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 4165	. . . . . . . . . 10 |- j C_ ~P U. j
18		pweq 3556	. . . . . . . . . 10 |- ( b = U. j -> ~P b = ~P U. j )
19	17, 18	sseqtrrid 3188	. . . . . . . . 9 |- ( b = U. j -> j C_ ~P b )
20		velpw 3560	. . . . . . . . 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 2522	. . . . . 6 |- { j e. Top | b = U. j } C_ ~P ~P b
24	15, 23	ssexi 4114	. . . . 5 |- { j e. Top | b = U. j } e. _V
25	12, 13, 24	fvmpt3i 5560	. . . 4 |- ( B e. _V -> (TopOn ` B ) = { j e. Top | B = U. j } )
26	25	eleq2d 2234	. . 3 |- ( B e. _V -> ( J e. (TopOn ` B ) <-> J e. { j e. Top | B = U. j } ) )
27		unieq 3792	. . . . 5 |- ( j = J -> U. j = U. J )
28	27	eqeq2d 2176	. . . 4 |- ( j = J -> ( B = U. j <-> B = U. J ) )
29	28	elrab 2877	. . 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 1342   e. wcel 2135   {crab 2446   _Vcvv 2721   C_ wss 3111   ~Pcpw 3553   U.cuni 3783   dom cdm 4598   Rel wrel 4603   Fun wfun 5176   ` cfv 5182   Topctop 12536  TopOnctopon 12549

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 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-13 2137  ax-14 2138  ax-ext 2146  ax-sep 4094  ax-pow 4147  ax-pr 4181  ax-un 4405

This theorem depends on definitions:  df-bi 116  df-3an 969  df-tru 1345  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ral 2447  df-rex 2448  df-rab 2451  df-v 2723  df-sbc 2947  df-un 3115  df-in 3117  df-ss 3124  df-pw 3555  df-sn 3576  df-pr 3577  df-op 3579  df-uni 3784  df-br 3977  df-opab 4038  df-mpt 4039  df-id 4265  df-xp 4604  df-rel 4605  df-cnv 4606  df-co 4607  df-dm 4608  df-iota 5147  df-fun 5184  df-fv 5190  df-topon 12550

This theorem is referenced by:  topontop  12553  toponuni  12554  toptopon  12557  toponcom  12566  istps2  12572  tgtopon  12607  distopon  12628  epttop  12631  resttopon  12712  resttopon2  12719  txtopon  12803