Theorem ustneism 21230

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 4115	. . . 4 |- ( A e. X -> { A } =/= (/) )
2	1	adantl 468	. . 3 |- ( ( V C_ ( X X. X ) /\ A e. X ) -> { A } =/= (/) )
3		xpco 5393	. . 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 5271	. . . . 5 |- `' ( { A } X. ( V " { A } ) ) = ( ( V " { A } ) X. { A } )
6		ressn 5389	. . . . . . 7 |- ( V |` { A } ) = ( { A } X. ( V " { A } ) )
7	6	cnveqi 5026	. . . . . 6 |- `' ( V |` { A } ) = `' ( { A } X. ( V " { A } ) )
8		resss 5145	. . . . . . 7 |- ( V |` { A } ) C_ V
9		cnvss 5024	. . . . . . 7 |- ( ( V |` { A } ) C_ V -> `' ( V |` { A } ) C_ `' V )
10	8, 9	ax-mp 5	. . . . . 6 |- `' ( V |` { A } ) C_ `' V
11	7, 10	eqsstr3i 3496	. . . . 5 |- `' ( { A } X. ( V " { A } ) ) C_ `' V
12	5, 11	eqsstr3i 3496	. . . 4 |- ( ( V " { A } ) X. { A } ) C_ `' V
13		coss2 5008	. . . 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 3496	. . . 4 |- ( { A } X. ( V " { A } ) ) C_ V
16		coss1 5007	. . . 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 3475	. 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 3500	1 |- ( ( V C_ ( X X. X ) /\ A e. X ) -> ( ( V " { A } ) X. ( V " { A } ) ) C_ ( V o. `' V ) )

Syntax hints:   -> wi 4   /\ wa 371   = wceq 1438   e. wcel 1869   =/= wne 2619   C_ wss 3437   (/)c0 3762   {csn 3997   X. cxp 4849   `'ccnv 4850   |` cres 4853   "cima 4854   o. ccom 4855

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-sep 4544  ax-nul 4553  ax-pr 4658

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-ral 2781  df-rex 2782  df-rab 2785  df-v 3084  df-sbc 3301  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3763  df-if 3911  df-sn 3998  df-pr 4000  df-op 4004  df-br 4422  df-opab 4481  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-rn 4862  df-res 4863  df-ima 4864

This theorem is referenced by:  neipcfilu  21303