Theorem ustdiag 22012

Description: The diagonal set is included in any entourage, i.e. any point is V -close to itself. Condition U_I of [BourbakiTop1] p. II.1. (Contributed by Thierry Arnoux, 2-Dec-2017.)

Assertion

Ref	Expression
ustdiag	|- ( ( U e. (UnifOn ` X ) /\ V e. U ) -> ( _I |` X ) C_ V )

Proof of Theorem ustdiag

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

Step	Hyp	Ref	Expression
1		elfvex 6221	. . . . . . 7 |- ( U e. (UnifOn ` X ) -> X e. _V )
2		isust 22007	. . . . . . 7 |- ( 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	. . . . . 6 |- ( 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 256	. . . . 5 |- ( 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 1075	. . . 4 |- ( 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 3626	. . . . . . . 8 |- ( v = V -> ( v C_ w <-> V C_ w ) )
7	6	imbi1d 331	. . . . . . 7 |- ( v = V -> ( ( v C_ w -> w e. U ) <-> ( V C_ w -> w e. U ) ) )
8	7	ralbidv 2986	. . . . . 6 |- ( 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 3807	. . . . . . . 8 |- ( v = V -> ( v i^i w ) = ( V i^i w ) )
10	9	eleq1d 2686	. . . . . . 7 |- ( v = V -> ( ( v i^i w ) e. U <-> ( V i^i w ) e. U ) )
11	10	ralbidv 2986	. . . . . 6 |- ( v = V -> ( A. w e. U ( v i^i w ) e. U <-> A. w e. U ( V i^i w ) e. U ) )
12		sseq2 3627	. . . . . . 7 |- ( v = V -> ( ( _I |` X ) C_ v <-> ( _I |` X ) C_ V ) )
13		cnveq 5296	. . . . . . . 8 |- ( v = V -> `' v = `' V )
14	13	eleq1d 2686	. . . . . . 7 |- ( v = V -> ( `' v e. U <-> `' V e. U ) )
15		sseq2 3627	. . . . . . . 8 |- ( v = V -> ( ( w o. w ) C_ v <-> ( w o. w ) C_ V ) )
16	15	rexbidv 3052	. . . . . . 7 |- ( 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 1399	. . . . . 6 |- ( 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 1399	. . . . 5 |- ( 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 3305	. . . 4 |- ( 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 486	. . 3 |- ( ( 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	simp3d 1075	. 2 |- ( ( U e. (UnifOn ` X ) /\ V e. U ) -> ( ( _I |` X ) C_ V /\ `' V e. U /\ E. w e. U ( w o. w ) C_ V ) )
22	21	simp1d 1073	1 |- ( ( U e. (UnifOn ` X ) /\ V e. U ) -> ( _I |` X ) C_ V )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   _Vcvv 3200   i^i cin 3573   C_ wss 3574   ~Pcpw 4158   _I cid 5023   X. cxp 5112   `'ccnv 5113   |` cres 5116   o. ccom 5118   ` cfv 5888  UnifOncust 22003

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-csb 3534  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-res 5126  df-iota 5851  df-fun 5890  df-fv 5896  df-ust 22004

This theorem is referenced by:  ustssco  22018  ustref  22022  ustelimasn  22026  trust  22033  ustuqtop3  22047