Theorem toponsspwpwg 12814

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 2741	. . 3 |- ( A e. V -> A e. _V )
2		rabssab 3235	. . . . . 6 |- { y e. Top | A = U. y } C_ { y | A = U. y }
3		eqcom 2172	. . . . . . 7 |- ( A = U. y <-> U. y = A )
4	3	abbii 2286	. . . . . 6 |- { y | A = U. y } = { y | U. y = A }
5	2, 4	sseqtri 3181	. . . . 5 |- { y e. Top | A = U. y } C_ { y | U. y = A }
6		pwpwssunieq 3961	. . . . 5 |- { y | U. y = A } C_ ~P ~P A
7	5, 6	sstri 3156	. . . 4 |- { y e. Top | A = U. y } C_ ~P ~P A
8		pwexg 4166	. . . . 5 |- ( A e. V -> ~P A e. _V )
9	8	pwexd 4167	. . . 4 |- ( A e. V -> ~P ~P A e. _V )
10		ssexg 4128	. . . 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 412	. . 3 |- ( A e. V -> { y e. Top | A = U. y } e. _V )
12		eqeq1 2177	. . . . 5 |- ( x = A -> ( x = U. y <-> A = U. y ) )
13	12	rabbidv 2719	. . . 4 |- ( x = A -> { y e. Top | x = U. y } = { y e. Top | A = U. y } )
14		df-topon 12803	. . . 4 |- TopOn = ( x e. _V |-> { y e. Top | x = U. y } )
15	13, 14	fvmptg 5572	. . 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 409	. 2 |- ( A e. V -> (TopOn ` A ) = { y e. Top | A = U. y } )
17	16, 7	eqsstrdi 3199	1 |- ( A e. V -> (TopOn ` A ) C_ ~P ~P A )

Syntax hints:   -> wi 4   = wceq 1348   e. wcel 2141   {cab 2156   {crab 2452   _Vcvv 2730   C_ wss 3121   ~Pcpw 3566   U.cuni 3796   ` cfv 5198   Topctop 12789  TopOnctopon 12802

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-iota 5160  df-fun 5200  df-fv 5206  df-topon 12803

This theorem is referenced by: (None)