Theorem ussval 22063

Description: The uniform structure on uniform space W. This proof uses a trick with fvprc 6185 to avoid requiring W to be a set. (Contributed by Thierry Arnoux, 3-Dec-2017.)

Hypotheses

Ref	Expression
ussval.1	|- B = ( Base ` W )
ussval.2	|- U = ( UnifSet ` W )

Assertion

Ref	Expression
ussval	|- ( U ↾t ( B X. B ) ) = (UnifSt ` W )

Proof of Theorem ussval

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6191	. . . . 5 |- ( w = W -> ( UnifSet ` w ) = ( UnifSet ` W ) )
2		fveq2 6191	. . . . . 6 |- ( w = W -> ( Base ` w ) = ( Base ` W ) )
3	2	sqxpeqd 5141	. . . . 5 |- ( w = W -> ( ( Base ` w ) X. ( Base ` w ) ) = ( ( Base ` W ) X. ( Base ` W ) ) )
4	1, 3	oveq12d 6668	. . . 4 |- ( w = W -> ( ( UnifSet ` w ) ↾t ( ( Base ` w ) X. ( Base ` w ) ) ) = ( ( UnifSet ` W ) ↾t ( ( Base ` W ) X. ( Base ` W ) ) ) )
5		df-uss 22060	. . . 4 |- UnifSt = ( w e. _V |-> ( ( UnifSet ` w ) ↾t ( ( Base ` w ) X. ( Base ` w ) ) ) )
6		ovex 6678	. . . 4 |- ( ( UnifSet ` W ) ↾t ( ( Base ` W ) X. ( Base ` W ) ) ) e. _V
7	4, 5, 6	fvmpt 6282	. . 3 |- ( W e. _V -> (UnifSt ` W ) = ( ( UnifSet ` W ) ↾t ( ( Base ` W ) X. ( Base ` W ) ) ) )
8		ussval.2	. . . 4 |- U = ( UnifSet ` W )
9		ussval.1	. . . . 5 |- B = ( Base ` W )
10	9, 9	xpeq12i 5137	. . . 4 |- ( B X. B ) = ( ( Base ` W ) X. ( Base ` W ) )
11	8, 10	oveq12i 6662	. . 3 |- ( U ↾t ( B X. B ) ) = ( ( UnifSet ` W ) ↾t ( ( Base ` W ) X. ( Base ` W ) ) )
12	7, 11	syl6reqr 2675	. 2 |- ( W e. _V -> ( U ↾t ( B X. B ) ) = (UnifSt ` W ) )
13		0rest 16090	. . 3 |- ( (/) ↾t ( B X. B ) ) = (/)
14		fvprc 6185	. . . . 5 |- ( -. W e. _V -> ( UnifSet ` W ) = (/) )
15	8, 14	syl5eq 2668	. . . 4 |- ( -. W e. _V -> U = (/) )
16	15	oveq1d 6665	. . 3 |- ( -. W e. _V -> ( U ↾t ( B X. B ) ) = ( (/) ↾t ( B X. B ) ) )
17		fvprc 6185	. . 3 |- ( -. W e. _V -> (UnifSt ` W ) = (/) )
18	13, 16, 17	3eqtr4a 2682	. 2 |- ( -. W e. _V -> ( U ↾t ( B X. B ) ) = (UnifSt ` W ) )
19	12, 18	pm2.61i 176	1 |- ( U ↾t ( B X. B ) ) = (UnifSt ` W )

Syntax hints:   -. wn 3   = wceq 1483   e. wcel 1990   _Vcvv 3200   (/)c0 3915   X. cxp 5112   ` cfv 5888  (class class class)co 6650   Basecbs 15857   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

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-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-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  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-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-rest 16083  df-uss 22060

This theorem is referenced by:  ussid  22064  ressuss  22067