Theorem ustneism 22027

Description: For a point A in X, ( V " { A } ) is small enough in ( V o. `' V ). This proposition actually does not require any axiom of the definition of uniform structures. (Contributed by Thierry Arnoux, 18-Nov-2017.)

Assertion

Ref	Expression
ustneism	|- ( ( V C_ ( X X. X ) /\ A e. X ) -> ( ( V " { A } ) X. ( V " { A } ) ) C_ ( V o. `' V ) )

Proof of Theorem ustneism

Step	Hyp	Ref	Expression
1		snnzg 4308	. . . 4 |- ( A e. X -> { A } =/= (/) )
2	1	adantl 482	. . 3 |- ( ( V C_ ( X X. X ) /\ A e. X ) -> { A } =/= (/) )
3		xpco 5675	. . 3 |- ( { A } =/= (/) -> ( ( { A } X. ( V " { A } ) ) o. ( ( V " { A } ) X. { A } ) ) = ( ( V " { A } ) X. ( V " { A } ) ) )
4	2, 3	syl 17	. 2 |- ( ( V C_ ( X X. X ) /\ A e. X ) -> ( ( { A } X. ( V " { A } ) ) o. ( ( V " { A } ) X. { A } ) ) = ( ( V " { A } ) X. ( V " { A } ) ) )
5		cnvxp 5551	. . . . 5 |- `' ( { A } X. ( V " { A } ) ) = ( ( V " { A } ) X. { A } )
6		ressn 5671	. . . . . . 7 |- ( V |` { A } ) = ( { A } X. ( V " { A } ) )
7	6	cnveqi 5297	. . . . . 6 |- `' ( V |` { A } ) = `' ( { A } X. ( V " { A } ) )
8		resss 5422	. . . . . . 7 |- ( V |` { A } ) C_ V
9		cnvss 5294	. . . . . . 7 |- ( ( V |` { A } ) C_ V -> `' ( V |` { A } ) C_ `' V )
10	8, 9	ax-mp 5	. . . . . 6 |- `' ( V |` { A } ) C_ `' V
11	7, 10	eqsstr3i 3636	. . . . 5 |- `' ( { A } X. ( V " { A } ) ) C_ `' V
12	5, 11	eqsstr3i 3636	. . . 4 |- ( ( V " { A } ) X. { A } ) C_ `' V
13		coss2 5278	. . . 4 |- ( ( ( V " { A } ) X. { A } ) C_ `' V -> ( ( { A } X. ( V " { A } ) ) o. ( ( V " { A } ) X. { A } ) ) C_ ( ( { A } X. ( V " { A } ) ) o. `' V ) )
14	12, 13	mp1i 13	. . 3 |- ( ( V C_ ( X X. X ) /\ A e. X ) -> ( ( { A } X. ( V " { A } ) ) o. ( ( V " { A } ) X. { A } ) ) C_ ( ( { A } X. ( V " { A } ) ) o. `' V ) )
15	6, 8	eqsstr3i 3636	. . . 4 |- ( { A } X. ( V " { A } ) ) C_ V
16		coss1 5277	. . . 4 |- ( ( { A } X. ( V " { A } ) ) C_ V -> ( ( { A } X. ( V " { A } ) ) o. `' V ) C_ ( V o. `' V ) )
17	15, 16	mp1i 13	. . 3 |- ( ( V C_ ( X X. X ) /\ A e. X ) -> ( ( { A } X. ( V " { A } ) ) o. `' V ) C_ ( V o. `' V ) )
18	14, 17	sstrd 3613	. 2 |- ( ( V C_ ( X X. X ) /\ A e. X ) -> ( ( { A } X. ( V " { A } ) ) o. ( ( V " { A } ) X. { A } ) ) C_ ( V o. `' V ) )
19	4, 18	eqsstr3d 3640	1 |- ( ( V C_ ( X X. X ) /\ A e. X ) -> ( ( V " { A } ) X. ( V " { A } ) ) C_ ( V o. `' V ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   C_ wss 3574   (/)c0 3915   {csn 4177   X. cxp 5112   `'ccnv 5113   |` cres 5116   "cima 5117   o. ccom 5118

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

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-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127

This theorem is referenced by:  neipcfilu  22100