Theorem ustincl 20442

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 5891	. . . . . . . 8 |- ( U e. (UnifOn ` X ) -> X e. _V )
2		isust 20438	. . . . . . . 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 16	. . . . . . 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 241	. . . . . 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 1010	. . . . 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 3525	. . . . . . . . 9 |- ( v = V -> ( v C_ w <-> V C_ w ) )
7	6	imbi1d 317	. . . . . . . 8 |- ( v = V -> ( ( v C_ w -> w e. U ) <-> ( V C_ w -> w e. U ) ) )
8	7	ralbidv 2903	. . . . . . 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 3693	. . . . . . . . 9 |- ( v = V -> ( v i^i w ) = ( V i^i w ) )
10	9	eleq1d 2536	. . . . . . . 8 |- ( v = V -> ( ( v i^i w ) e. U <-> ( V i^i w ) e. U ) )
11	10	ralbidv 2903	. . . . . . 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 3526	. . . . . . . 8 |- ( v = V -> ( ( _I |` X ) C_ v <-> ( _I |` X ) C_ V ) )
13		cnveq 5174	. . . . . . . . 9 |- ( v = V -> `' v = `' V )
14	13	eleq1d 2536	. . . . . . . 8 |- ( v = V -> ( `' v e. U <-> `' V e. U ) )
15		sseq2 3526	. . . . . . . . 9 |- ( v = V -> ( ( w o. w ) C_ v <-> ( w o. w ) C_ V ) )
16	15	rexbidv 2973	. . . . . . . 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 1299	. . . . . . 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 1299	. . . . . 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 3210	. . . . 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 469	. . . 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 1009	. . 3 |- ( ( U e. (UnifOn ` X ) /\ V e. U ) -> A. w e. U ( V i^i w ) e. U )
22		ineq2 3694	. . . . 5 |- ( w = W -> ( V i^i w ) = ( V i^i W ) )
23	22	eleq1d 2536	. . . 4 |- ( w = W -> ( ( V i^i w ) e. U <-> ( V i^i W ) e. U ) )
24	23	rspcv 3210	. . 3 |- ( W e. U -> ( A. w e. U ( V i^i w ) e. U -> ( V i^i W ) e. U ) )
25	21, 24	mpan9 469	. 2 |- ( ( ( U e. (UnifOn ` X ) /\ V e. U ) /\ W e. U ) -> ( V i^i W ) e. U )
26	25	3impa 1191	1 |- ( ( U e. (UnifOn ` X ) /\ V e. U /\ W e. U ) -> ( V i^i W ) e. U )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 973   = wceq 1379   e. wcel 1767   A.wral 2814   E.wrex 2815   _Vcvv 3113   i^i cin 3475   C_ wss 3476   ~Pcpw 4010   _I cid 4790   X. cxp 4997   `'ccnv 4998   |` cres 5001   o. ccom 5003   ` cfv 5586  UnifOncust 20434

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-res 5011  df-iota 5549  df-fun 5588  df-fv 5594  df-ust 20435

This theorem is referenced by:  ustexsym  20450  trust  20464  utoptop  20469  restutopopn  20473  ustuqtop2  20477