Theorem istopon 13653

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 13652	. . . . 5 |- Fun TopOn
2		funrel 5235	. . . . 5 |- ( Fun TopOn -> Rel TopOn )
3	1, 2	ax-mp 5	. . . 4 |- Rel TopOn
4		relelfvdm 5549	. . . 4 |- ( ( Rel TopOn /\ J e. (TopOn ` B ) ) -> B e. dom TopOn )
5	3, 4	mpan 424	. . 3 |- ( J e. (TopOn ` B ) -> B e. dom TopOn )
6	5	elexd 2752	. 2 |- ( J e. (TopOn ` B ) -> B e. _V )
7		uniexg 4441	. . . 4 |- ( J e. Top -> U. J e. _V )
8		eleq1 2240	. . . 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 2184	. . . . . 6 |- ( b = B -> ( b = U. j <-> B = U. j ) )
12	11	rabbidv 2728	. . . . 5 |- ( b = B -> { j e. Top | b = U. j } = { j e. Top | B = U. j } )
13		df-topon 13651	. . . . 5 |- TopOn = ( b e. _V |-> { j e. Top | b = U. j } )
14		vpwex 4181	. . . . . . 7 |- ~P b e. _V
15	14	pwex 4185	. . . . . 6 |- ~P ~P b e. _V
16		rabss 3234	. . . . . . 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 4194	. . . . . . . . . 10 |- j C_ ~P U. j
18		pweq 3580	. . . . . . . . . 10 |- ( b = U. j -> ~P b = ~P U. j )
19	17, 18	sseqtrrid 3208	. . . . . . . . 9 |- ( b = U. j -> j C_ ~P b )
20		velpw 3584	. . . . . . . . 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 2535	. . . . . 6 |- { j e. Top | b = U. j } C_ ~P ~P b
24	15, 23	ssexi 4143	. . . . 5 |- { j e. Top | b = U. j } e. _V
25	12, 13, 24	fvmpt3i 5599	. . . 4 |- ( B e. _V -> (TopOn ` B ) = { j e. Top | B = U. j } )
26	25	eleq2d 2247	. . 3 |- ( B e. _V -> ( J e. (TopOn ` B ) <-> J e. { j e. Top | B = U. j } ) )
27		unieq 3820	. . . . 5 |- ( j = J -> U. j = U. J )
28	27	eqeq2d 2189	. . . 4 |- ( j = J -> ( B = U. j <-> B = U. J ) )
29	28	elrab 2895	. . 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 704	1 |- ( J e. (TopOn ` B ) <-> ( J e. Top /\ B = U. J ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1353   e. wcel 2148   {crab 2459   _Vcvv 2739   C_ wss 3131   ~Pcpw 3577   U.cuni 3811   dom cdm 4628   Rel wrel 4633   Fun wfun 5212   ` cfv 5218   Topctop 13637  TopOnctopon 13650

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2741  df-sbc 2965  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-iota 5180  df-fun 5220  df-fv 5226  df-topon 13651

This theorem is referenced by:  topontop  13654  toponuni  13655  toptopon  13658  toponcom  13667  istps2  13673  tgtopon  13706  distopon  13727  epttop  13730  resttopon  13811  resttopon2  13818  txtopon  13902