Theorem ustincl 21270

Description: A uniform structure is closed under finite intersection. Condition F_II of [BourbakiTop1] p. I.36. (Contributed by Thierry Arnoux, 30-Nov-2017.)

Assertion

Ref	Expression
ustincl	|- ( ( U e. (UnifOn ` X ) /\ V e. U /\ W e. U ) -> ( V i^i W ) e. U )

Proof of Theorem ustincl

Dummy variables v w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elfvex 5914	. . . . . . . 8 |- ( U e. (UnifOn ` X ) -> X e. _V )
2		isust 21266	. . . . . . . 8 |- ( X e. _V -> ( U e. (UnifOn ` X ) <-> ( U C_ ~P ( X X. X ) /\ ( X X. X ) e. U /\ A. v e. U ( A. w e. ~P ( X X. X ) ( v C_ w -> w e. U ) /\ A. w e. U ( v i^i w ) e. U /\ ( ( _I |` X ) C_ v /\ `' v e. U /\ E. w e. U ( w o. w ) C_ v ) ) ) ) )
3	1, 2	syl 17	. . . . . . 7 |- ( U e. (UnifOn ` X ) -> ( U e. (UnifOn ` X ) <-> ( U C_ ~P ( X X. X ) /\ ( X X. X ) e. U /\ A. v e. U ( A. w e. ~P ( X X. X ) ( v C_ w -> w e. U ) /\ A. w e. U ( v i^i w ) e. U /\ ( ( _I |` X ) C_ v /\ `' v e. U /\ E. w e. U ( w o. w ) C_ v ) ) ) ) )
4	3	ibi 249	. . . . . 6 |- ( U e. (UnifOn ` X ) -> ( U C_ ~P ( X X. X ) /\ ( X X. X ) e. U /\ A. v e. U ( A. w e. ~P ( X X. X ) ( v C_ w -> w e. U ) /\ A. w e. U ( v i^i w ) e. U /\ ( ( _I |` X ) C_ v /\ `' v e. U /\ E. w e. U ( w o. w ) C_ v ) ) ) )
5	4	simp3d 1028	. . . . 5 |- ( U e. (UnifOn ` X ) -> A. v e. U ( A. w e. ~P ( X X. X ) ( v C_ w -> w e. U ) /\ A. w e. U ( v i^i w ) e. U /\ ( ( _I |` X ) C_ v /\ `' v e. U /\ E. w e. U ( w o. w ) C_ v ) ) )
6		sseq1 3464	. . . . . . . . 9 |- ( v = V -> ( v C_ w <-> V C_ w ) )
7	6	imbi1d 323	. . . . . . . 8 |- ( v = V -> ( ( v C_ w -> w e. U ) <-> ( V C_ w -> w e. U ) ) )
8	7	ralbidv 2838	. . . . . . 7 |- ( v = V -> ( A. w e. ~P ( X X. X ) ( v C_ w -> w e. U ) <-> A. w e. ~P ( X X. X ) ( V C_ w -> w e. U ) ) )
9		ineq1 3638	. . . . . . . . 9 |- ( v = V -> ( v i^i w ) = ( V i^i w ) )
10	9	eleq1d 2523	. . . . . . . 8 |- ( v = V -> ( ( v i^i w ) e. U <-> ( V i^i w ) e. U ) )
11	10	ralbidv 2838	. . . . . . 7 |- ( v = V -> ( A. w e. U ( v i^i w ) e. U <-> A. w e. U ( V i^i w ) e. U ) )
12		sseq2 3465	. . . . . . . 8 |- ( v = V -> ( ( _I |` X ) C_ v <-> ( _I |` X ) C_ V ) )
13		cnveq 5026	. . . . . . . . 9 |- ( v = V -> `' v = `' V )
14	13	eleq1d 2523	. . . . . . . 8 |- ( v = V -> ( `' v e. U <-> `' V e. U ) )
15		sseq2 3465	. . . . . . . . 9 |- ( v = V -> ( ( w o. w ) C_ v <-> ( w o. w ) C_ V ) )
16	15	rexbidv 2912	. . . . . . . 8 |- ( v = V -> ( E. w e. U ( w o. w ) C_ v <-> E. w e. U ( w o. w ) C_ V ) )
17	12, 14, 16	3anbi123d 1348	. . . . . . 7 |- ( v = V -> ( ( ( _I |` X ) C_ v /\ `' v e. U /\ E. w e. U ( w o. w ) C_ v ) <-> ( ( _I |` X ) C_ V /\ `' V e. U /\ E. w e. U ( w o. w ) C_ V ) ) )
18	8, 11, 17	3anbi123d 1348	. . . . . 6 |- ( v = V -> ( ( A. w e. ~P ( X X. X ) ( v C_ w -> w e. U ) /\ A. w e. U ( v i^i w ) e. U /\ ( ( _I |` X ) C_ v /\ `' v e. U /\ E. w e. U ( w o. w ) C_ v ) ) <-> ( A. w e. ~P ( X X. X ) ( V C_ w -> w e. U ) /\ A. w e. U ( V i^i w ) e. U /\ ( ( _I |` X ) C_ V /\ `' V e. U /\ E. w e. U ( w o. w ) C_ V ) ) ) )
19	18	rspcv 3157	. . . . 5 |- ( V e. U -> ( A. v e. U ( A. w e. ~P ( X X. X ) ( v C_ w -> w e. U ) /\ A. w e. U ( v i^i w ) e. U /\ ( ( _I |` X ) C_ v /\ `' v e. U /\ E. w e. U ( w o. w ) C_ v ) ) -> ( A. w e. ~P ( X X. X ) ( V C_ w -> w e. U ) /\ A. w e. U ( V i^i w ) e. U /\ ( ( _I |` X ) C_ V /\ `' V e. U /\ E. w e. U ( w o. w ) C_ V ) ) ) )
20	5, 19	mpan9 476	. . . 4 |- ( ( U e. (UnifOn ` X ) /\ V e. U ) -> ( A. w e. ~P ( X X. X ) ( V C_ w -> w e. U ) /\ A. w e. U ( V i^i w ) e. U /\ ( ( _I |` X ) C_ V /\ `' V e. U /\ E. w e. U ( w o. w ) C_ V ) ) )
21	20	simp2d 1027	. . 3 |- ( ( U e. (UnifOn ` X ) /\ V e. U ) -> A. w e. U ( V i^i w ) e. U )
22		ineq2 3639	. . . . 5 |- ( w = W -> ( V i^i w ) = ( V i^i W ) )
23	22	eleq1d 2523	. . . 4 |- ( w = W -> ( ( V i^i w ) e. U <-> ( V i^i W ) e. U ) )
24	23	rspcv 3157	. . 3 |- ( W e. U -> ( A. w e. U ( V i^i w ) e. U -> ( V i^i W ) e. U ) )
25	21, 24	mpan9 476	. 2 |- ( ( ( U e. (UnifOn ` X ) /\ V e. U ) /\ W e. U ) -> ( V i^i W ) e. U )
26	25	3impa 1210	1 |- ( ( U e. (UnifOn ` X ) /\ V e. U /\ W e. U ) -> ( V i^i W ) e. U )

Syntax hints:   -> wi 4   <-> wb 189   /\ wa 375   /\ w3a 991   = wceq 1454   e. wcel 1897   A.wral 2748   E.wrex 2749   _Vcvv 3056   i^i cin 3414   C_ wss 3415   ~Pcpw 3962   _I cid 4762   X. cxp 4850   `'ccnv 4851   |` cres 4854   o. ccom 4856   ` cfv 5600  UnifOncust 21262

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-8 1899  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-sep 4538  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6609

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-ral 2753  df-rex 2754  df-rab 2757  df-v 3058  df-sbc 3279  df-csb 3375  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-nul 3743  df-if 3893  df-pw 3964  df-sn 3980  df-pr 3982  df-op 3986  df-uni 4212  df-br 4416  df-opab 4475  df-mpt 4476  df-id 4767  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-res 4864  df-iota 5564  df-fun 5602  df-fv 5608  df-ust 21263

This theorem is referenced by:  ustexsym  21278  trust  21292  utoptop  21297  restutopopn  21301  ustuqtop2  21305