Theorem toponsspwpw 20726

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

Assertion

Ref	Expression
toponsspwpw	|- (TopOn ` A ) C_ ~P ~P A

Proof of Theorem toponsspwpw

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

Step	Hyp	Ref	Expression
1		rabssab 3690	. . . . . . 7 |- { y e. Top | A = U. y } C_ { y | A = U. y }
2		eqcom 2629	. . . . . . . 8 |- ( A = U. y <-> U. y = A )
3	2	abbii 2739	. . . . . . 7 |- { y | A = U. y } = { y | U. y = A }
4	1, 3	sseqtri 3637	. . . . . 6 |- { y e. Top | A = U. y } C_ { y | U. y = A }
5		pwpwssunieq 4615	. . . . . 6 |- { y | U. y = A } C_ ~P ~P A
6	4, 5	sstri 3612	. . . . 5 |- { y e. Top | A = U. y } C_ ~P ~P A
7		pwexg 4850	. . . . . 6 |- ( A e. _V -> ~P A e. _V )
8		pwexg 4850	. . . . . 6 |- ( ~P A e. _V -> ~P ~P A e. _V )
9	7, 8	syl 17	. . . . 5 |- ( A e. _V -> ~P ~P A e. _V )
10		ssexg 4804	. . . . 5 |- ( ( { y e. Top | A = U. y } C_ ~P ~P A /\ ~P ~P A e. _V ) -> { y e. Top | A = U. y } e. _V )
11	6, 9, 10	sylancr 695	. . . 4 |- ( A e. _V -> { y e. Top | A = U. y } e. _V )
12		eqeq1 2626	. . . . . 6 |- ( x = A -> ( x = U. y <-> A = U. y ) )
13	12	rabbidv 3189	. . . . 5 |- ( x = A -> { y e. Top | x = U. y } = { y e. Top | A = U. y } )
14		df-topon 20716	. . . . 5 |- TopOn = ( x e. _V |-> { y e. Top | x = U. y } )
15	13, 14	fvmptg 6280	. . . 4 |- ( ( A e. _V /\ { y e. Top | A = U. y } e. _V ) -> (TopOn ` A ) = { y e. Top | A = U. y } )
16	11, 15	mpdan 702	. . 3 |- ( A e. _V -> (TopOn ` A ) = { y e. Top | A = U. y } )
17	16, 6	syl6eqss 3655	. 2 |- ( A e. _V -> (TopOn ` A ) C_ ~P ~P A )
18		fvprc 6185	. . 3 |- ( -. A e. _V -> (TopOn ` A ) = (/) )
19		0ss 3972	. . 3 |- (/) C_ ~P ~P A
20	18, 19	syl6eqss 3655	. 2 |- ( -. A e. _V -> (TopOn ` A ) C_ ~P ~P A )
21	17, 20	pm2.61i 176	1 |- (TopOn ` A ) C_ ~P ~P A

Syntax hints:   -. wn 3   = wceq 1483   e. wcel 1990   {cab 2608   {crab 2916   _Vcvv 3200   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   U.cuni 4436   ` cfv 5888   Topctop 20698  TopOnctopon 20715

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-topon 20716

This theorem is referenced by: (None)