Theorem topmeet 32359

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 32358	. . . 4 |- ( ( X e. V /\ S C_ (TopOn ` X ) ) -> ( ~P X i^i |^| S ) e. (TopOn ` X ) )
2		inss2 3834	. . . . . . 7 |- ( ~P X i^i |^| S ) C_ |^| S
3		intss1 4492	. . . . . . 7 |- ( j e. S -> |^| S C_ j )
4	2, 3	syl5ss 3614	. . . . . 6 |- ( j e. S -> ( ~P X i^i |^| S ) C_ j )
5	4	rgen 2922	. . . . 5 |- A. j e. S ( ~P X i^i |^| S ) C_ j
6		sseq1 3626	. . . . . . 7 |- ( k = ( ~P X i^i |^| S ) -> ( k C_ j <-> ( ~P X i^i |^| S ) C_ j ) )
7	6	ralbidv 2986	. . . . . 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 3363	. . . . 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 954	. . . 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 224	. . 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 4467	. . 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 20719	. . . . . . . . 9 |- ( k e. (TopOn ` X ) -> X = U. k )
14		eqimss2 3658	. . . . . . . . 9 |- ( X = U. k -> U. k C_ X )
15	13, 14	syl 17	. . . . . . . 8 |- ( k e. (TopOn ` X ) -> U. k C_ X )
16		sspwuni 4611	. . . . . . . 8 |- ( k C_ ~P X <-> U. k C_ X )
17	15, 16	sylibr 224	. . . . . . 7 |- ( k e. (TopOn ` X ) -> k C_ ~P X )
18	17	3ad2ant2 1083	. . . . . 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 1063	. . . . . . 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 4493	. . . . . . 7 |- ( k C_ |^| S <-> A. j e. S k C_ j )
21	19, 20	sylibr 224	. . . . . 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 3837	. . . . 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 4165	. . . . 5 |- ( k e. ~P ( ~P X i^i |^| S ) <-> k C_ ( ~P X i^i |^| S ) )
24	22, 23	sylibr 224	. . . 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 3682	. . 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 4611	. . 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 208	. 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 3620	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 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   i^i cin 3573   C_ wss 3574   ~Pcpw 4158   U.cuni 4436   |^|cint 4475   ` cfv 5888  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  ax-un 6949

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-ne 2795  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-int 4476  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-mre 16246  df-top 20699  df-topon 20716

This theorem is referenced by: (None)