Theorem istopon 14600

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 14599	. . . . 5 |- Fun TopOn
2		funrel 5307	. . . . 5 |- ( Fun TopOn -> Rel TopOn )
3	1, 2	ax-mp 5	. . . 4 |- Rel TopOn
4		relelfvdm 5631	. . . 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 2790	. 2 |- ( J e. (TopOn ` B ) -> B e. _V )
7		uniexg 4504	. . . 4 |- ( J e. Top -> U. J e. _V )
8		eleq1 2270	. . . 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 2214	. . . . . 6 |- ( b = B -> ( b = U. j <-> B = U. j ) )
12	11	rabbidv 2765	. . . . 5 |- ( b = B -> { j e. Top | b = U. j } = { j e. Top | B = U. j } )
13		df-topon 14598	. . . . 5 |- TopOn = ( b e. _V |-> { j e. Top | b = U. j } )
14		vpwex 4239	. . . . . . 7 |- ~P b e. _V
15	14	pwex 4243	. . . . . 6 |- ~P ~P b e. _V
16		rabss 3278	. . . . . . 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 4252	. . . . . . . . . 10 |- j C_ ~P U. j
18		pweq 3629	. . . . . . . . . 10 |- ( b = U. j -> ~P b = ~P U. j )
19	17, 18	sseqtrrid 3252	. . . . . . . . 9 |- ( b = U. j -> j C_ ~P b )
20		velpw 3633	. . . . . . . . 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 2566	. . . . . 6 |- { j e. Top | b = U. j } C_ ~P ~P b
24	15, 23	ssexi 4198	. . . . 5 |- { j e. Top | b = U. j } e. _V
25	12, 13, 24	fvmpt3i 5682	. . . 4 |- ( B e. _V -> (TopOn ` B ) = { j e. Top | B = U. j } )
26	25	eleq2d 2277	. . 3 |- ( B e. _V -> ( J e. (TopOn ` B ) <-> J e. { j e. Top | B = U. j } ) )
27		unieq 3873	. . . . 5 |- ( j = J -> U. j = U. J )
28	27	eqeq2d 2219	. . . 4 |- ( j = J -> ( B = U. j <-> B = U. J ) )
29	28	elrab 2936	. . 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 706	1 |- ( J e. (TopOn ` B ) <-> ( J e. Top /\ B = U. J ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1373   e. wcel 2178   {crab 2490   _Vcvv 2776   C_ wss 3174   ~Pcpw 3626   U.cuni 3864   dom cdm 4693   Rel wrel 4698   Fun wfun 5284   ` cfv 5290   Topctop 14584  TopOnctopon 14597

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-rab 2495  df-v 2778  df-sbc 3006  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-iota 5251  df-fun 5292  df-fv 5298  df-topon 14598

This theorem is referenced by:  topontop  14601  toponuni  14602  toptopon  14605  toponcom  14614  istps2  14620  tgtopon  14653  distopon  14674  epttop  14677  resttopon  14758  resttopon2  14765  txtopon  14849