Theorem ustssco 22018

Description: In an uniform structure, any entourage V is a subset of its composition with itself. (Contributed by Thierry Arnoux, 5-Jan-2018.)

Assertion

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

Proof of Theorem ustssco

Step	Hyp	Ref	Expression
1		ssun1 3776	. . . 4 |- V C_ ( V u. ( V o. V ) )
2		coires1 5653	. . . . . 6 |- ( V o. ( _I |` X ) ) = ( V |` X )
3		ustrel 22015	. . . . . . 7 |- ( ( U e. (UnifOn ` X ) /\ V e. U ) -> Rel V )
4		ustssxp 22008	. . . . . . . . 9 |- ( ( U e. (UnifOn ` X ) /\ V e. U ) -> V C_ ( X X. X ) )
5		dmss 5323	. . . . . . . . 9 |- ( V C_ ( X X. X ) -> dom V C_ dom ( X X. X ) )
6	4, 5	syl 17	. . . . . . . 8 |- ( ( U e. (UnifOn ` X ) /\ V e. U ) -> dom V C_ dom ( X X. X ) )
7		dmxpid 5345	. . . . . . . 8 |- dom ( X X. X ) = X
8	6, 7	syl6sseq 3651	. . . . . . 7 |- ( ( U e. (UnifOn ` X ) /\ V e. U ) -> dom V C_ X )
9		relssres 5437	. . . . . . 7 |- ( ( Rel V /\ dom V C_ X ) -> ( V |` X ) = V )
10	3, 8, 9	syl2anc 693	. . . . . 6 |- ( ( U e. (UnifOn ` X ) /\ V e. U ) -> ( V |` X ) = V )
11	2, 10	syl5eq 2668	. . . . 5 |- ( ( U e. (UnifOn ` X ) /\ V e. U ) -> ( V o. ( _I |` X ) ) = V )
12	11	uneq1d 3766	. . . 4 |- ( ( U e. (UnifOn ` X ) /\ V e. U ) -> ( ( V o. ( _I |` X ) ) u. ( V o. V ) ) = ( V u. ( V o. V ) ) )
13	1, 12	syl5sseqr 3654	. . 3 |- ( ( U e. (UnifOn ` X ) /\ V e. U ) -> V C_ ( ( V o. ( _I |` X ) ) u. ( V o. V ) ) )
14		coundi 5636	. . 3 |- ( V o. ( ( _I |` X ) u. V ) ) = ( ( V o. ( _I |` X ) ) u. ( V o. V ) )
15	13, 14	syl6sseqr 3652	. 2 |- ( ( U e. (UnifOn ` X ) /\ V e. U ) -> V C_ ( V o. ( ( _I |` X ) u. V ) ) )
16		ustdiag 22012	. . . 4 |- ( ( U e. (UnifOn ` X ) /\ V e. U ) -> ( _I |` X ) C_ V )
17		ssequn1 3783	. . . 4 |- ( ( _I |` X ) C_ V <-> ( ( _I |` X ) u. V ) = V )
18	16, 17	sylib 208	. . 3 |- ( ( U e. (UnifOn ` X ) /\ V e. U ) -> ( ( _I |` X ) u. V ) = V )
19	18	coeq2d 5284	. 2 |- ( ( U e. (UnifOn ` X ) /\ V e. U ) -> ( V o. ( ( _I |` X ) u. V ) ) = ( V o. V ) )
20	15, 19	sseqtrd 3641	1 |- ( ( U e. (UnifOn ` X ) /\ V e. U ) -> V C_ ( V o. V ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   u. cun 3572   C_ wss 3574   _I cid 5023   X. cxp 5112   dom cdm 5114   |` cres 5116   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:  ustexsym  22019  ustex3sym  22021