Theorem topmeet 31091

Description: Two equivalent formulations of the meet of a collection of topologies. (Contributed by Jeff Hankins, 4-Oct-2009.) (Proof shortened by Mario Carneiro, 12-Sep-2015.)

Assertion

Ref	Expression
topmeet	|- ( ( X e. V /\ S C_ (TopOn ` X ) ) -> ( ~P X i^i |^| S ) = U. { k e. (TopOn ` X ) | A. j e. S k C_ j } )

Distinct variable groups:   j, k, S   j, V, k   j, X, k

Proof of Theorem topmeet

Step	Hyp	Ref	Expression
1		topmtcl 31090	. . . 4 |- ( ( X e. V /\ S C_ (TopOn ` X ) ) -> ( ~P X i^i |^| S ) e. (TopOn ` X ) )
2		inss2 3644	. . . . . . 7 |- ( ~P X i^i |^| S ) C_ |^| S
3		intss1 4241	. . . . . . 7 |- ( j e. S -> |^| S C_ j )
4	2, 3	syl5ss 3429	. . . . . 6 |- ( j e. S -> ( ~P X i^i |^| S ) C_ j )
5	4	rgen 2766	. . . . 5 |- A. j e. S ( ~P X i^i |^| S ) C_ j
6		sseq1 3439	. . . . . . 7 |- ( k = ( ~P X i^i |^| S ) -> ( k C_ j <-> ( ~P X i^i |^| S ) C_ j ) )
7	6	ralbidv 2829	. . . . . 6 |- ( k = ( ~P X i^i |^| S ) -> ( A. j e. S k C_ j <-> A. j e. S ( ~P X i^i |^| S ) C_ j ) )
8	7	elrab 3184	. . . . 5 |- ( ( ~P X i^i |^| S ) e. { k e. (TopOn ` X ) | A. j e. S k C_ j } <-> ( ( ~P X i^i |^| S ) e. (TopOn ` X ) /\ A. j e. S ( ~P X i^i |^| S ) C_ j ) )
9	5, 8	mpbiran2 933	. . . 4 |- ( ( ~P X i^i |^| S ) e. { k e. (TopOn ` X ) | A. j e. S k C_ j } <-> ( ~P X i^i |^| S ) e. (TopOn ` X ) )
10	1, 9	sylibr 217	. . 3 |- ( ( X e. V /\ S C_ (TopOn ` X ) ) -> ( ~P X i^i |^| S ) e. { k e. (TopOn ` X ) | A. j e. S k C_ j } )
11		elssuni 4219	. . 3 |- ( ( ~P X i^i |^| S ) e. { k e. (TopOn ` X ) | A. j e. S k C_ j } -> ( ~P X i^i |^| S ) C_ U. { k e. (TopOn ` X ) | A. j e. S k C_ j } )
12	10, 11	syl 17	. 2 |- ( ( X e. V /\ S C_ (TopOn ` X ) ) -> ( ~P X i^i |^| S ) C_ U. { k e. (TopOn ` X ) | A. j e. S k C_ j } )
13		toponuni 20019	. . . . . . . . 9 |- ( k e. (TopOn ` X ) -> X = U. k )
14		eqimss2 3471	. . . . . . . . 9 |- ( X = U. k -> U. k C_ X )
15	13, 14	syl 17	. . . . . . . 8 |- ( k e. (TopOn ` X ) -> U. k C_ X )
16		sspwuni 4360	. . . . . . . 8 |- ( k C_ ~P X <-> U. k C_ X )
17	15, 16	sylibr 217	. . . . . . 7 |- ( k e. (TopOn ` X ) -> k C_ ~P X )
18	17	3ad2ant2 1052	. . . . . 6 |- ( ( ( X e. V /\ S C_ (TopOn ` X ) ) /\ k e. (TopOn ` X ) /\ A. j e. S k C_ j ) -> k C_ ~P X )
19		simp3 1032	. . . . . . 7 |- ( ( ( X e. V /\ S C_ (TopOn ` X ) ) /\ k e. (TopOn ` X ) /\ A. j e. S k C_ j ) -> A. j e. S k C_ j )
20		ssint 4242	. . . . . . 7 |- ( k C_ |^| S <-> A. j e. S k C_ j )
21	19, 20	sylibr 217	. . . . . 6 |- ( ( ( X e. V /\ S C_ (TopOn ` X ) ) /\ k e. (TopOn ` X ) /\ A. j e. S k C_ j ) -> k C_ |^| S )
22	18, 21	ssind 3647	. . . . 5 |- ( ( ( X e. V /\ S C_ (TopOn ` X ) ) /\ k e. (TopOn ` X ) /\ A. j e. S k C_ j ) -> k C_ ( ~P X i^i |^| S ) )
23		selpw 3949	. . . . 5 |- ( k e. ~P ( ~P X i^i |^| S ) <-> k C_ ( ~P X i^i |^| S ) )
24	22, 23	sylibr 217	. . . 4 |- ( ( ( X e. V /\ S C_ (TopOn ` X ) ) /\ k e. (TopOn ` X ) /\ A. j e. S k C_ j ) -> k e. ~P ( ~P X i^i |^| S ) )
25	24	rabssdv 3495	. . 3 |- ( ( X e. V /\ S C_ (TopOn ` X ) ) -> { k e. (TopOn ` X ) | A. j e. S k C_ j } C_ ~P ( ~P X i^i |^| S ) )
26		sspwuni 4360	. . 3 |- ( { k e. (TopOn ` X ) | A. j e. S k C_ j } C_ ~P ( ~P X i^i |^| S ) <-> U. { k e. (TopOn ` X ) | A. j e. S k C_ j } C_ ( ~P X i^i |^| S ) )
27	25, 26	sylib 201	. 2 |- ( ( X e. V /\ S C_ (TopOn ` X ) ) -> U. { k e. (TopOn ` X ) | A. j e. S k C_ j } C_ ( ~P X i^i |^| S ) )
28	12, 27	eqssd 3435	1 |- ( ( X e. V /\ S C_ (TopOn ` X ) ) -> ( ~P X i^i |^| S ) = U. { k e. (TopOn ` X ) | A. j e. S k C_ j } )

Syntax hints:   -> wi 4   /\ wa 376   /\ w3a 1007   = wceq 1452   e. wcel 1904   A.wral 2756   {crab 2760   i^i cin 3389   C_ wss 3390   ~Pcpw 3942   U.cuni 4190   |^|cint 4226   ` cfv 5589  TopOnctopon 19995

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-int 4227  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-iota 5553  df-fun 5591  df-fv 5597  df-mre 15570  df-top 19998  df-topon 20000

This theorem is referenced by: (None)