Theorem ussid 22064

Description: In case the base of the UnifSt element of the uniform space is the base of its element structure, then UnifSt does not restrict it further. (Contributed by Thierry Arnoux, 4-Dec-2017.)

Hypotheses

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

Assertion

Ref	Expression
ussid	|- ( ( B X. B ) = U. U -> U = (UnifSt ` W ) )

Proof of Theorem ussid

Step	Hyp	Ref	Expression
1		oveq2 6658	. . 3 |- ( ( B X. B ) = U. U -> ( U ↾t ( B X. B ) ) = ( U ↾t U. U ) )
2		id 22	. . . . . 6 |- ( ( B X. B ) = U. U -> ( B X. B ) = U. U )
3		ussval.1	. . . . . . . 8 |- B = ( Base ` W )
4		fvex 6201	. . . . . . . 8 |- ( Base ` W ) e. _V
5	3, 4	eqeltri 2697	. . . . . . 7 |- B e. _V
6	5, 5	xpex 6962	. . . . . 6 |- ( B X. B ) e. _V
7	2, 6	syl6eqelr 2710	. . . . 5 |- ( ( B X. B ) = U. U -> U. U e. _V )
8		uniexb 6973	. . . . 5 |- ( U e. _V <-> U. U e. _V )
9	7, 8	sylibr 224	. . . 4 |- ( ( B X. B ) = U. U -> U e. _V )
10		eqid 2622	. . . . 5 |- U. U = U. U
11	10	restid 16094	. . . 4 |- ( U e. _V -> ( U ↾t U. U ) = U )
12	9, 11	syl 17	. . 3 |- ( ( B X. B ) = U. U -> ( U ↾t U. U ) = U )
13	1, 12	eqtr2d 2657	. 2 |- ( ( B X. B ) = U. U -> U = ( U ↾t ( B X. B ) ) )
14		ussval.2	. . 3 |- U = ( UnifSet ` W )
15	3, 14	ussval 22063	. 2 |- ( U ↾t ( B X. B ) ) = (UnifSt ` W )
16	13, 15	syl6eq 2672	1 |- ( ( B X. B ) = U. U -> U = (UnifSt ` W ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   _Vcvv 3200   U.cuni 4436   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-pw 4160  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:  tususs  22074  cnflduss  23152