Theorem ressuss 22067

Description: Value of the uniform structure of a restricted space. (Contributed by Thierry Arnoux, 12-Dec-2017.)

Assertion

Ref	Expression
ressuss	|- ( A e. V -> (UnifSt ` ( W ↾s A ) ) = ( (UnifSt ` W ) ↾t ( A X. A ) ) )

Proof of Theorem ressuss

Step	Hyp	Ref	Expression
1		eqid 2622	. . . . 5 |- ( Base ` W ) = ( Base ` W )
2		eqid 2622	. . . . 5 |- ( UnifSet ` W ) = ( UnifSet ` W )
3	1, 2	ussval 22063	. . . 4 |- ( ( UnifSet ` W ) ↾t ( ( Base ` W ) X. ( Base ` W ) ) ) = (UnifSt ` W )
4	3	oveq1i 6660	. . 3 |- ( ( ( UnifSet ` W ) ↾t ( ( Base ` W ) X. ( Base ` W ) ) ) ↾t ( A X. A ) ) = ( (UnifSt ` W ) ↾t ( A X. A ) )
5		fvex 6201	. . . . 5 |- ( UnifSet ` W ) e. _V
6	5	a1i 11	. . . 4 |- ( A e. V -> ( UnifSet ` W ) e. _V )
7		fvex 6201	. . . . . 6 |- ( Base ` W ) e. _V
8	7, 7	xpex 6962	. . . . 5 |- ( ( Base ` W ) X. ( Base ` W ) ) e. _V
9	8	a1i 11	. . . 4 |- ( A e. V -> ( ( Base ` W ) X. ( Base ` W ) ) e. _V )
10		sqxpexg 6963	. . . 4 |- ( A e. V -> ( A X. A ) e. _V )
11		restco 20968	. . . 4 |- ( ( ( UnifSet ` W ) e. _V /\ ( ( Base ` W ) X. ( Base ` W ) ) e. _V /\ ( A X. A ) e. _V ) -> ( ( ( UnifSet ` W ) ↾t ( ( Base ` W ) X. ( Base ` W ) ) ) ↾t ( A X. A ) ) = ( ( UnifSet ` W ) ↾t ( ( ( Base ` W ) X. ( Base ` W ) ) i^i ( A X. A ) ) ) )
12	6, 9, 10, 11	syl3anc 1326	. . 3 |- ( A e. V -> ( ( ( UnifSet ` W ) ↾t ( ( Base ` W ) X. ( Base ` W ) ) ) ↾t ( A X. A ) ) = ( ( UnifSet ` W ) ↾t ( ( ( Base ` W ) X. ( Base ` W ) ) i^i ( A X. A ) ) ) )
13	4, 12	syl5eqr 2670	. 2 |- ( A e. V -> ( (UnifSt ` W ) ↾t ( A X. A ) ) = ( ( UnifSet ` W ) ↾t ( ( ( Base ` W ) X. ( Base ` W ) ) i^i ( A X. A ) ) ) )
14		inxp 5254	. . . . 5 |- ( ( ( Base ` W ) X. ( Base ` W ) ) i^i ( A X. A ) ) = ( ( ( Base ` W ) i^i A ) X. ( ( Base ` W ) i^i A ) )
15		incom 3805	. . . . . . 7 |- ( A i^i ( Base ` W ) ) = ( ( Base ` W ) i^i A )
16		eqid 2622	. . . . . . . 8 |- ( W ↾s A ) = ( W ↾s A )
17	16, 1	ressbas 15930	. . . . . . 7 |- ( A e. V -> ( A i^i ( Base ` W ) ) = ( Base ` ( W ↾s A ) ) )
18	15, 17	syl5eqr 2670	. . . . . 6 |- ( A e. V -> ( ( Base ` W ) i^i A ) = ( Base ` ( W ↾s A ) ) )
19	18	sqxpeqd 5141	. . . . 5 |- ( A e. V -> ( ( ( Base ` W ) i^i A ) X. ( ( Base ` W ) i^i A ) ) = ( ( Base ` ( W ↾s A ) ) X. ( Base ` ( W ↾s A ) ) ) )
20	14, 19	syl5eq 2668	. . . 4 |- ( A e. V -> ( ( ( Base ` W ) X. ( Base ` W ) ) i^i ( A X. A ) ) = ( ( Base ` ( W ↾s A ) ) X. ( Base ` ( W ↾s A ) ) ) )
21	20	oveq2d 6666	. . 3 |- ( A e. V -> ( ( UnifSet ` W ) ↾t ( ( ( Base ` W ) X. ( Base ` W ) ) i^i ( A X. A ) ) ) = ( ( UnifSet ` W ) ↾t ( ( Base ` ( W ↾s A ) ) X. ( Base ` ( W ↾s A ) ) ) ) )
22	16, 2	ressunif 22066	. . . 4 |- ( A e. V -> ( UnifSet ` W ) = ( UnifSet ` ( W ↾s A ) ) )
23	22	oveq1d 6665	. . 3 |- ( A e. V -> ( ( UnifSet ` W ) ↾t ( ( Base ` ( W ↾s A ) ) X. ( Base ` ( W ↾s A ) ) ) ) = ( ( UnifSet ` ( W ↾s A ) ) ↾t ( ( Base ` ( W ↾s A ) ) X. ( Base ` ( W ↾s A ) ) ) ) )
24		eqid 2622	. . . . 5 |- ( Base ` ( W ↾s A ) ) = ( Base ` ( W ↾s A ) )
25		eqid 2622	. . . . 5 |- ( UnifSet ` ( W ↾s A ) ) = ( UnifSet ` ( W ↾s A ) )
26	24, 25	ussval 22063	. . . 4 |- ( ( UnifSet ` ( W ↾s A ) ) ↾t ( ( Base ` ( W ↾s A ) ) X. ( Base ` ( W ↾s A ) ) ) ) = (UnifSt ` ( W ↾s A ) )
27	26	a1i 11	. . 3 |- ( A e. V -> ( ( UnifSet ` ( W ↾s A ) ) ↾t ( ( Base ` ( W ↾s A ) ) X. ( Base ` ( W ↾s A ) ) ) ) = (UnifSt ` ( W ↾s A ) ) )
28	21, 23, 27	3eqtrd 2660	. 2 |- ( A e. V -> ( ( UnifSet ` W ) ↾t ( ( ( Base ` W ) X. ( Base ` W ) ) i^i ( A X. A ) ) ) = (UnifSt ` ( W ↾s A ) ) )
29	13, 28	eqtr2d 2657	1 |- ( A e. V -> (UnifSt ` ( W ↾s A ) ) = ( (UnifSt ` W ) ↾t ( A X. A ) ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   _Vcvv 3200   i^i cin 3573   X. cxp 5112   ` cfv 5888  (class class class)co 6650   Basecbs 15857   ↾_s cress 15858   UnifSetcunif 15951   ↾_t crest 16081  UnifStcuss 22057

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-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  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-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-unif 15965  df-rest 16083  df-uss 22060

This theorem is referenced by:  ressust  22068  ressusp  22069  ucnextcn  22108  reust  23169  qqhucn  30036