Theorem ustneism 19798

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 3992	. . . 4 |- ( A e. X -> { A } =/= (/) )
2	1	adantl 466	. . 3 |- ( ( V C_ ( X X. X ) /\ A e. X ) -> { A } =/= (/) )
3		xpco 5377	. . 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 5255	. . . . 5 |- `' ( { A } X. ( V " { A } ) ) = ( ( V " { A } ) X. { A } )
6		ressn 5373	. . . . . . 7 |- ( V |` { A } ) = ( { A } X. ( V " { A } ) )
7	6	cnveqi 5014	. . . . . 6 |- `' ( V |` { A } ) = `' ( { A } X. ( V " { A } ) )
8		resss 5134	. . . . . . 7 |- ( V |` { A } ) C_ V
9		cnvss 5012	. . . . . . 7 |- ( ( V |` { A } ) C_ V -> `' ( V |` { A } ) C_ `' V )
10	8, 9	ax-mp 5	. . . . . 6 |- `' ( V |` { A } ) C_ `' V
11	7, 10	eqsstr3i 3387	. . . . 5 |- `' ( { A } X. ( V " { A } ) ) C_ `' V
12	5, 11	eqsstr3i 3387	. . . 4 |- ( ( V " { A } ) X. { A } ) C_ `' V
13		coss2 4996	. . . 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 3387	. . . 4 |- ( { A } X. ( V " { A } ) ) C_ V
16		coss1 4995	. . . 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 3366	. 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 3391	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 1369   e. wcel 1756   =/= wne 2606   C_ wss 3328   (/)c0 3637   {csn 3877   X. cxp 4838   `'ccnv 4839   |` cres 4842   "cima 4843   o. ccom 4844

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pr 4531

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-op 3884  df-br 4293  df-opab 4351  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853

This theorem is referenced by:  neipcfilu  19871