Theorem toponsspwpwg 13071

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 2746	. . 3 |- ( A e. V -> A e. _V )
2		rabssab 3241	. . . . . 6 |- { y e. Top | A = U. y } C_ { y | A = U. y }
3		eqcom 2177	. . . . . . 7 |- ( A = U. y <-> U. y = A )
4	3	abbii 2291	. . . . . 6 |- { y | A = U. y } = { y | U. y = A }
5	2, 4	sseqtri 3187	. . . . 5 |- { y e. Top | A = U. y } C_ { y | U. y = A }
6		pwpwssunieq 3970	. . . . 5 |- { y | U. y = A } C_ ~P ~P A
7	5, 6	sstri 3162	. . . 4 |- { y e. Top | A = U. y } C_ ~P ~P A
8		pwexg 4175	. . . . 5 |- ( A e. V -> ~P A e. _V )
9	8	pwexd 4176	. . . 4 |- ( A e. V -> ~P ~P A e. _V )
10		ssexg 4137	. . . 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 2182	. . . . 5 |- ( x = A -> ( x = U. y <-> A = U. y ) )
13	12	rabbidv 2724	. . . 4 |- ( x = A -> { y e. Top | x = U. y } = { y e. Top | A = U. y } )
14		df-topon 13060	. . . 4 |- TopOn = ( x e. _V |-> { y e. Top | x = U. y } )
15	13, 14	fvmptg 5584	. . 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 3205	1 |- ( A e. V -> (TopOn ` A ) C_ ~P ~P A )

Syntax hints:   -> wi 4   = wceq 1353   e. wcel 2146   {cab 2161   {crab 2457   _Vcvv 2735   C_ wss 3127   ~Pcpw 3572   U.cuni 3805   ` cfv 5208   Topctop 13046  TopOnctopon 13059

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 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-pow 4169  ax-pr 4203

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ral 2458  df-rex 2459  df-rab 2462  df-v 2737  df-sbc 2961  df-un 3131  df-in 3133  df-ss 3140  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-br 3999  df-opab 4060  df-mpt 4061  df-id 4287  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-iota 5170  df-fun 5210  df-fv 5216  df-topon 13060

This theorem is referenced by: (None)