Theorem toponsspwpwg 15013

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 2827	. . 3 |- ( A e. V -> A e. _V )
2		rabssab 3331	. . . . . 6 |- { y e. Top | A = U. y } C_ { y | A = U. y }
3		eqcom 2236	. . . . . . 7 |- ( A = U. y <-> U. y = A )
4	3	abbii 2350	. . . . . 6 |- { y | A = U. y } = { y | U. y = A }
5	2, 4	sseqtri 3276	. . . . 5 |- { y e. Top | A = U. y } C_ { y | U. y = A }
6		pwpwssunieq 4085	. . . . 5 |- { y | U. y = A } C_ ~P ~P A
7	5, 6	sstri 3251	. . . 4 |- { y e. Top | A = U. y } C_ ~P ~P A
8		pwexg 4298	. . . . 5 |- ( A e. V -> ~P A e. _V )
9	8	pwexd 4299	. . . 4 |- ( A e. V -> ~P ~P A e. _V )
10		ssexg 4254	. . . 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 2241	. . . . 5 |- ( x = A -> ( x = U. y <-> A = U. y ) )
13	12	rabbidv 2804	. . . 4 |- ( x = A -> { y e. Top | x = U. y } = { y e. Top | A = U. y } )
14		df-topon 15002	. . . 4 |- TopOn = ( x e. _V |-> { y e. Top | x = U. y } )
15	13, 14	fvmptg 5758	. . 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 3294	1 |- ( A e. V -> (TopOn ` A ) C_ ~P ~P A )

Syntax hints:   -> wi 4   = wceq 1398   e. wcel 2205   {cab 2220   {crab 2526   _Vcvv 2815   C_ wss 3214   ~Pcpw 3674   U.cuni 3919   ` cfv 5357   Topctop 14988  TopOnctopon 15001

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-sbc 3046  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-iota 5317  df-fun 5359  df-fv 5365  df-topon 15002

This theorem is referenced by: (None)