Theorem ustneism 20892

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 4133	. . . 4 |- ( A e. X -> { A } =/= (/) )
2	1	adantl 464	. . 3 |- ( ( V C_ ( X X. X ) /\ A e. X ) -> { A } =/= (/) )
3		xpco 5530	. . 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 5409	. . . . 5 |- `' ( { A } X. ( V " { A } ) ) = ( ( V " { A } ) X. { A } )
6		ressn 5526	. . . . . . 7 |- ( V |` { A } ) = ( { A } X. ( V " { A } ) )
7	6	cnveqi 5166	. . . . . 6 |- `' ( V |` { A } ) = `' ( { A } X. ( V " { A } ) )
8		resss 5285	. . . . . . 7 |- ( V |` { A } ) C_ V
9		cnvss 5164	. . . . . . 7 |- ( ( V |` { A } ) C_ V -> `' ( V |` { A } ) C_ `' V )
10	8, 9	ax-mp 5	. . . . . 6 |- `' ( V |` { A } ) C_ `' V
11	7, 10	eqsstr3i 3520	. . . . 5 |- `' ( { A } X. ( V " { A } ) ) C_ `' V
12	5, 11	eqsstr3i 3520	. . . 4 |- ( ( V " { A } ) X. { A } ) C_ `' V
13		coss2 5148	. . . 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 3520	. . . 4 |- ( { A } X. ( V " { A } ) ) C_ V
16		coss1 5147	. . . 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 3499	. 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 3524	1 |- ( ( V C_ ( X X. X ) /\ A e. X ) -> ( ( V " { A } ) X. ( V " { A } ) ) C_ ( V o. `' V ) )

Syntax hints:   -> wi 4   /\ wa 367   = wceq 1398   e. wcel 1823   =/= wne 2649   C_ wss 3461   (/)c0 3783   {csn 4016   X. cxp 4986   `'ccnv 4987   |` cres 4990   "cima 4991   o. ccom 4992

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-br 4440  df-opab 4498  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001

This theorem is referenced by:  neipcfilu  20965