Theorem toponsspwpwg 14745

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 2814	. . 3 |- ( A e. V -> A e. _V )
2		rabssab 3315	. . . . . 6 |- { y e. Top | A = U. y } C_ { y | A = U. y }
3		eqcom 2233	. . . . . . 7 |- ( A = U. y <-> U. y = A )
4	3	abbii 2347	. . . . . 6 |- { y | A = U. y } = { y | U. y = A }
5	2, 4	sseqtri 3261	. . . . 5 |- { y e. Top | A = U. y } C_ { y | U. y = A }
6		pwpwssunieq 4059	. . . . 5 |- { y | U. y = A } C_ ~P ~P A
7	5, 6	sstri 3236	. . . 4 |- { y e. Top | A = U. y } C_ ~P ~P A
8		pwexg 4270	. . . . 5 |- ( A e. V -> ~P A e. _V )
9	8	pwexd 4271	. . . 4 |- ( A e. V -> ~P ~P A e. _V )
10		ssexg 4228	. . . 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 2238	. . . . 5 |- ( x = A -> ( x = U. y <-> A = U. y ) )
13	12	rabbidv 2791	. . . 4 |- ( x = A -> { y e. Top | x = U. y } = { y e. Top | A = U. y } )
14		df-topon 14734	. . . 4 |- TopOn = ( x e. _V |-> { y e. Top | x = U. y } )
15	13, 14	fvmptg 5722	. . 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 3279	1 |- ( A e. V -> (TopOn ` A ) C_ ~P ~P A )

Syntax hints:   -> wi 4   = wceq 1397   e. wcel 2202   {cab 2217   {crab 2514   _Vcvv 2802   C_ wss 3200   ~Pcpw 3652   U.cuni 3893   ` cfv 5326   Topctop 14720  TopOnctopon 14733

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-sbc 3032  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-iota 5286  df-fun 5328  df-fv 5334  df-topon 14734

This theorem is referenced by: (None)