Theorem ustneism 20477

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 4144	. . . 4 |- ( A e. X -> { A } =/= (/) )
2	1	adantl 466	. . 3 |- ( ( V C_ ( X X. X ) /\ A e. X ) -> { A } =/= (/) )
3		xpco 5546	. . 3 |- ( { A } =/= (/) -> ( ( { A } X. ( V " { A } ) ) o. ( ( V " { A } ) X. { A } ) ) = ( ( V " { A } ) X. ( V " { A } ) ) )
4	2, 3	syl 16	. 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 5423	. . . . 5 |- `' ( { A } X. ( V " { A } ) ) = ( ( V " { A } ) X. { A } )
6		ressn 5542	. . . . . . 7 |- ( V |` { A } ) = ( { A } X. ( V " { A } ) )
7	6	cnveqi 5176	. . . . . 6 |- `' ( V |` { A } ) = `' ( { A } X. ( V " { A } ) )
8		resss 5296	. . . . . . 7 |- ( V |` { A } ) C_ V
9		cnvss 5174	. . . . . . 7 |- ( ( V |` { A } ) C_ V -> `' ( V |` { A } ) C_ `' V )
10	8, 9	ax-mp 5	. . . . . 6 |- `' ( V |` { A } ) C_ `' V
11	7, 10	eqsstr3i 3535	. . . . 5 |- `' ( { A } X. ( V " { A } ) ) C_ `' V
12	5, 11	eqsstr3i 3535	. . . 4 |- ( ( V " { A } ) X. { A } ) C_ `' V
13		coss2 5158	. . . 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 12	. . 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 3535	. . . 4 |- ( { A } X. ( V " { A } ) ) C_ V
16		coss1 5157	. . . 4 |- ( ( { A } X. ( V " { A } ) ) C_ V -> ( ( { A } X. ( V " { A } ) ) o. `' V ) C_ ( V o. `' V ) )
17	15, 16	mp1i 12	. . 3 |- ( ( V C_ ( X X. X ) /\ A e. X ) -> ( ( { A } X. ( V " { A } ) ) o. `' V ) C_ ( V o. `' V ) )
18	14, 17	sstrd 3514	. 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 3539	1 |- ( ( V C_ ( X X. X ) /\ A e. X ) -> ( ( V " { A } ) X. ( V " { A } ) ) C_ ( V o. `' V ) )

Syntax hints:   -> wi 4   /\ wa 369   = wceq 1379   e. wcel 1767   =/= wne 2662   C_ wss 3476   (/)c0 3785   {csn 4027   X. cxp 4997   `'ccnv 4998   |` cres 5001   "cima 5002   o. ccom 5003

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-opab 4506  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012

This theorem is referenced by:  neipcfilu  20550