Theorem ustexsym 22019

Description: In an uniform structure, for any entourage V, there exists a smaller symmetrical entourage. (Contributed by Thierry Arnoux, 4-Jan-2018.)

Assertion

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

Distinct variable groups:   w, U   w, V

Allowed substitution hint:   X( w)

Proof of Theorem ustexsym

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simplll 798	. . . 4 |- ( ( ( ( U e. (UnifOn ` X ) /\ V e. U ) /\ x e. U ) /\ ( x o. x ) C_ V ) -> U e. (UnifOn ` X ) )
2		simplr 792	. . . . 5 |- ( ( ( ( U e. (UnifOn ` X ) /\ V e. U ) /\ x e. U ) /\ ( x o. x ) C_ V ) -> x e. U )
3		ustinvel 22013	. . . . 5 |- ( ( U e. (UnifOn ` X ) /\ x e. U ) -> `' x e. U )
4	1, 2, 3	syl2anc 693	. . . 4 |- ( ( ( ( U e. (UnifOn ` X ) /\ V e. U ) /\ x e. U ) /\ ( x o. x ) C_ V ) -> `' x e. U )
5		ustincl 22011	. . . 4 |- ( ( U e. (UnifOn ` X ) /\ `' x e. U /\ x e. U ) -> ( `' x i^i x ) e. U )
6	1, 4, 2, 5	syl3anc 1326	. . 3 |- ( ( ( ( U e. (UnifOn ` X ) /\ V e. U ) /\ x e. U ) /\ ( x o. x ) C_ V ) -> ( `' x i^i x ) e. U )
7		ustrel 22015	. . . . . . 7 |- ( ( U e. (UnifOn ` X ) /\ x e. U ) -> Rel x )
8		dfrel2 5583	. . . . . . 7 |- ( Rel x <-> `' `' x = x )
9	7, 8	sylib 208	. . . . . 6 |- ( ( U e. (UnifOn ` X ) /\ x e. U ) -> `' `' x = x )
10	9	ineq1d 3813	. . . . 5 |- ( ( U e. (UnifOn ` X ) /\ x e. U ) -> ( `' `' x i^i `' x ) = ( x i^i `' x ) )
11		cnvin 5540	. . . . 5 |- `' ( `' x i^i x ) = ( `' `' x i^i `' x )
12		incom 3805	. . . . 5 |- ( `' x i^i x ) = ( x i^i `' x )
13	10, 11, 12	3eqtr4g 2681	. . . 4 |- ( ( U e. (UnifOn ` X ) /\ x e. U ) -> `' ( `' x i^i x ) = ( `' x i^i x ) )
14	1, 2, 13	syl2anc 693	. . 3 |- ( ( ( ( U e. (UnifOn ` X ) /\ V e. U ) /\ x e. U ) /\ ( x o. x ) C_ V ) -> `' ( `' x i^i x ) = ( `' x i^i x ) )
15		inss2 3834	. . . 4 |- ( `' x i^i x ) C_ x
16		ustssco 22018	. . . . . 6 |- ( ( U e. (UnifOn ` X ) /\ x e. U ) -> x C_ ( x o. x ) )
17	1, 2, 16	syl2anc 693	. . . . 5 |- ( ( ( ( U e. (UnifOn ` X ) /\ V e. U ) /\ x e. U ) /\ ( x o. x ) C_ V ) -> x C_ ( x o. x ) )
18		simpr 477	. . . . 5 |- ( ( ( ( U e. (UnifOn ` X ) /\ V e. U ) /\ x e. U ) /\ ( x o. x ) C_ V ) -> ( x o. x ) C_ V )
19	17, 18	sstrd 3613	. . . 4 |- ( ( ( ( U e. (UnifOn ` X ) /\ V e. U ) /\ x e. U ) /\ ( x o. x ) C_ V ) -> x C_ V )
20	15, 19	syl5ss 3614	. . 3 |- ( ( ( ( U e. (UnifOn ` X ) /\ V e. U ) /\ x e. U ) /\ ( x o. x ) C_ V ) -> ( `' x i^i x ) C_ V )
21		cnveq 5296	. . . . . 6 |- ( w = ( `' x i^i x ) -> `' w = `' ( `' x i^i x ) )
22		id 22	. . . . . 6 |- ( w = ( `' x i^i x ) -> w = ( `' x i^i x ) )
23	21, 22	eqeq12d 2637	. . . . 5 |- ( w = ( `' x i^i x ) -> ( `' w = w <-> `' ( `' x i^i x ) = ( `' x i^i x ) ) )
24		sseq1 3626	. . . . 5 |- ( w = ( `' x i^i x ) -> ( w C_ V <-> ( `' x i^i x ) C_ V ) )
25	23, 24	anbi12d 747	. . . 4 |- ( w = ( `' x i^i x ) -> ( ( `' w = w /\ w C_ V ) <-> ( `' ( `' x i^i x ) = ( `' x i^i x ) /\ ( `' x i^i x ) C_ V ) ) )
26	25	rspcev 3309	. . 3 |- ( ( ( `' x i^i x ) e. U /\ ( `' ( `' x i^i x ) = ( `' x i^i x ) /\ ( `' x i^i x ) C_ V ) ) -> E. w e. U ( `' w = w /\ w C_ V ) )
27	6, 14, 20, 26	syl12anc 1324	. 2 |- ( ( ( ( U e. (UnifOn ` X ) /\ V e. U ) /\ x e. U ) /\ ( x o. x ) C_ V ) -> E. w e. U ( `' w = w /\ w C_ V ) )
28		ustexhalf 22014	. 2 |- ( ( U e. (UnifOn ` X ) /\ V e. U ) -> E. x e. U ( x o. x ) C_ V )
29	27, 28	r19.29a 3078	1 |- ( ( U e. (UnifOn ` X ) /\ V e. U ) -> E. w e. U ( `' w = w /\ w C_ V ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   i^i cin 3573   C_ wss 3574   `'ccnv 5113   o. ccom 5118   Rel wrel 5119   ` 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:  ustex2sym  22020  neipcfilu  22100