Theorem ustex2sym 22020

Description: In an uniform structure, for any entourage V, there exists a symmetrical entourage smaller than half V. (Contributed by Thierry Arnoux, 16-Jan-2018.)

Assertion

Ref	Expression
ustex2sym	|- ( ( U e. (UnifOn ` X ) /\ V e. U ) -> E. w e. U ( `' w = w /\ ( w o. w ) C_ V ) )

Distinct variable groups:   w, U   w, V   w, X

Proof of Theorem ustex2sym

Dummy variable v is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simplll 798	. . . 4 |- ( ( ( ( U e. (UnifOn ` X ) /\ V e. U ) /\ v e. U ) /\ ( v o. v ) C_ V ) -> U e. (UnifOn ` X ) )
2		simplr 792	. . . 4 |- ( ( ( ( U e. (UnifOn ` X ) /\ V e. U ) /\ v e. U ) /\ ( v o. v ) C_ V ) -> v e. U )
3		ustexsym 22019	. . . 4 |- ( ( U e. (UnifOn ` X ) /\ v e. U ) -> E. w e. U ( `' w = w /\ w C_ v ) )
4	1, 2, 3	syl2anc 693	. . 3 |- ( ( ( ( U e. (UnifOn ` X ) /\ V e. U ) /\ v e. U ) /\ ( v o. v ) C_ V ) -> E. w e. U ( `' w = w /\ w C_ v ) )
5		simprl 794	. . . . . 6 |- ( ( ( ( ( ( U e. (UnifOn ` X ) /\ V e. U ) /\ v e. U ) /\ ( v o. v ) C_ V ) /\ w e. U ) /\ ( `' w = w /\ w C_ v ) ) -> `' w = w )
6		coss1 5277	. . . . . . . . 9 |- ( w C_ v -> ( w o. w ) C_ ( v o. w ) )
7		coss2 5278	. . . . . . . . 9 |- ( w C_ v -> ( v o. w ) C_ ( v o. v ) )
8	6, 7	sstrd 3613	. . . . . . . 8 |- ( w C_ v -> ( w o. w ) C_ ( v o. v ) )
9	8	ad2antll 765	. . . . . . 7 |- ( ( ( ( ( ( U e. (UnifOn ` X ) /\ V e. U ) /\ v e. U ) /\ ( v o. v ) C_ V ) /\ w e. U ) /\ ( `' w = w /\ w C_ v ) ) -> ( w o. w ) C_ ( v o. v ) )
10		simpllr 799	. . . . . . 7 |- ( ( ( ( ( ( U e. (UnifOn ` X ) /\ V e. U ) /\ v e. U ) /\ ( v o. v ) C_ V ) /\ w e. U ) /\ ( `' w = w /\ w C_ v ) ) -> ( v o. v ) C_ V )
11	9, 10	sstrd 3613	. . . . . 6 |- ( ( ( ( ( ( U e. (UnifOn ` X ) /\ V e. U ) /\ v e. U ) /\ ( v o. v ) C_ V ) /\ w e. U ) /\ ( `' w = w /\ w C_ v ) ) -> ( w o. w ) C_ V )
12	5, 11	jca 554	. . . . 5 |- ( ( ( ( ( ( U e. (UnifOn ` X ) /\ V e. U ) /\ v e. U ) /\ ( v o. v ) C_ V ) /\ w e. U ) /\ ( `' w = w /\ w C_ v ) ) -> ( `' w = w /\ ( w o. w ) C_ V ) )
13	12	ex 450	. . . 4 |- ( ( ( ( ( U e. (UnifOn ` X ) /\ V e. U ) /\ v e. U ) /\ ( v o. v ) C_ V ) /\ w e. U ) -> ( ( `' w = w /\ w C_ v ) -> ( `' w = w /\ ( w o. w ) C_ V ) ) )
14	13	reximdva 3017	. . 3 |- ( ( ( ( U e. (UnifOn ` X ) /\ V e. U ) /\ v e. U ) /\ ( v o. v ) C_ V ) -> ( E. w e. U ( `' w = w /\ w C_ v ) -> E. w e. U ( `' w = w /\ ( w o. w ) C_ V ) ) )
15	4, 14	mpd 15	. 2 |- ( ( ( ( U e. (UnifOn ` X ) /\ V e. U ) /\ v e. U ) /\ ( v o. v ) C_ V ) -> E. w e. U ( `' w = w /\ ( w o. w ) C_ V ) )
16		ustexhalf 22014	. 2 |- ( ( U e. (UnifOn ` X ) /\ V e. U ) -> E. v e. U ( v o. v ) C_ V )
17	15, 16	r19.29a 3078	1 |- ( ( U e. (UnifOn ` X ) /\ V e. U ) -> E. w e. U ( `' w = w /\ ( w o. w ) C_ V ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   C_ wss 3574   `'ccnv 5113   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-rn 5125  df-res 5126  df-iota 5851  df-fun 5890  df-fv 5896  df-ust 22004

This theorem is referenced by:  ustex3sym  22021