Theorem toponsspwpwg 14190

Description: The set of topologies on a set is included in the double power set of that set. (Contributed by BJ, 29-Apr-2021.) (Revised by Jim Kingdon, 16-Jan-2023.)

Assertion

Ref	Expression
toponsspwpwg	|- ( A e. V -> (TopOn ` A ) C_ ~P ~P A )

Proof of Theorem toponsspwpwg

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 2771	. . 3 |- ( A e. V -> A e. _V )
2		rabssab 3267	. . . . . 6 |- { y e. Top | A = U. y } C_ { y | A = U. y }
3		eqcom 2195	. . . . . . 7 |- ( A = U. y <-> U. y = A )
4	3	abbii 2309	. . . . . 6 |- { y | A = U. y } = { y | U. y = A }
5	2, 4	sseqtri 3213	. . . . 5 |- { y e. Top | A = U. y } C_ { y | U. y = A }
6		pwpwssunieq 4001	. . . . 5 |- { y | U. y = A } C_ ~P ~P A
7	5, 6	sstri 3188	. . . 4 |- { y e. Top | A = U. y } C_ ~P ~P A
8		pwexg 4209	. . . . 5 |- ( A e. V -> ~P A e. _V )
9	8	pwexd 4210	. . . 4 |- ( A e. V -> ~P ~P A e. _V )
10		ssexg 4168	. . . 4 |- ( ( { y e. Top | A = U. y } C_ ~P ~P A /\ ~P ~P A e. _V ) -> { y e. Top | A = U. y } e. _V )
11	7, 9, 10	sylancr 414	. . 3 |- ( A e. V -> { y e. Top | A = U. y } e. _V )
12		eqeq1 2200	. . . . 5 |- ( x = A -> ( x = U. y <-> A = U. y ) )
13	12	rabbidv 2749	. . . 4 |- ( x = A -> { y e. Top | x = U. y } = { y e. Top | A = U. y } )
14		df-topon 14179	. . . 4 |- TopOn = ( x e. _V |-> { y e. Top | x = U. y } )
15	13, 14	fvmptg 5633	. . 3 |- ( ( A e. _V /\ { y e. Top | A = U. y } e. _V ) -> (TopOn ` A ) = { y e. Top | A = U. y } )
16	1, 11, 15	syl2anc 411	. 2 |- ( A e. V -> (TopOn ` A ) = { y e. Top | A = U. y } )
17	16, 7	eqsstrdi 3231	1 |- ( A e. V -> (TopOn ` A ) C_ ~P ~P A )

Syntax hints:   -> wi 4   = wceq 1364   e. wcel 2164   {cab 2179   {crab 2476   _Vcvv 2760   C_ wss 3153   ~Pcpw 3601   U.cuni 3835   ` cfv 5254   Topctop 14165  TopOnctopon 14178

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-sbc 2986  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-iota 5215  df-fun 5256  df-fv 5262  df-topon 14179

This theorem is referenced by: (None)