Theorem tusval 22070

Description: The value of the uniform space mapping function. (Contributed by Thierry Arnoux, 5-Dec-2017.)

Assertion

Ref	Expression
tusval	|- ( U e. (UnifOn ` X ) -> (toUnifSp ` U ) = ( { <. ( Base ` ndx ) , dom U. U >. , <. ( UnifSet ` ndx ) , U >. } sSet <. (TopSet ` ndx ) , (unifTop ` U ) >. ) )

Proof of Theorem tusval

Dummy variable u is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-tus 22062	. . 3 |- toUnifSp = ( u e. U. ran UnifOn |-> ( { <. ( Base ` ndx ) , dom U. u >. , <. ( UnifSet ` ndx ) , u >. } sSet <. (TopSet ` ndx ) , (unifTop ` u ) >. ) )
2	1	a1i 11	. 2 |- ( U e. (UnifOn ` X ) -> toUnifSp = ( u e. U. ran UnifOn |-> ( { <. ( Base ` ndx ) , dom U. u >. , <. ( UnifSet ` ndx ) , u >. } sSet <. (TopSet ` ndx ) , (unifTop ` u ) >. ) ) )
3		simpr 477	. . . . . . 7 |- ( ( U e. (UnifOn ` X ) /\ u = U ) -> u = U )
4	3	unieqd 4446	. . . . . 6 |- ( ( U e. (UnifOn ` X ) /\ u = U ) -> U. u = U. U )
5	4	dmeqd 5326	. . . . 5 |- ( ( U e. (UnifOn ` X ) /\ u = U ) -> dom U. u = dom U. U )
6	5	opeq2d 4409	. . . 4 |- ( ( U e. (UnifOn ` X ) /\ u = U ) -> <. ( Base ` ndx ) , dom U. u >. = <. ( Base ` ndx ) , dom U. U >. )
7	3	opeq2d 4409	. . . 4 |- ( ( U e. (UnifOn ` X ) /\ u = U ) -> <. ( UnifSet ` ndx ) , u >. = <. ( UnifSet ` ndx ) , U >. )
8	6, 7	preq12d 4276	. . 3 |- ( ( U e. (UnifOn ` X ) /\ u = U ) -> { <. ( Base ` ndx ) , dom U. u >. , <. ( UnifSet ` ndx ) , u >. } = { <. ( Base ` ndx ) , dom U. U >. , <. ( UnifSet ` ndx ) , U >. } )
9	3	fveq2d 6195	. . . 4 |- ( ( U e. (UnifOn ` X ) /\ u = U ) -> (unifTop ` u ) = (unifTop ` U ) )
10	9	opeq2d 4409	. . 3 |- ( ( U e. (UnifOn ` X ) /\ u = U ) -> <. (TopSet ` ndx ) , (unifTop ` u ) >. = <. (TopSet ` ndx ) , (unifTop ` U ) >. )
11	8, 10	oveq12d 6668	. 2 |- ( ( U e. (UnifOn ` X ) /\ u = U ) -> ( { <. ( Base ` ndx ) , dom U. u >. , <. ( UnifSet ` ndx ) , u >. } sSet <. (TopSet ` ndx ) , (unifTop ` u ) >. ) = ( { <. ( Base ` ndx ) , dom U. U >. , <. ( UnifSet ` ndx ) , U >. } sSet <. (TopSet ` ndx ) , (unifTop ` U ) >. ) )
12		elrnust 22028	. 2 |- ( U e. (UnifOn ` X ) -> U e. U. ran UnifOn )
13		ovexd 6680	. 2 |- ( U e. (UnifOn ` X ) -> ( { <. ( Base ` ndx ) , dom U. U >. , <. ( UnifSet ` ndx ) , U >. } sSet <. (TopSet ` ndx ) , (unifTop ` U ) >. ) e. _V )
14	2, 11, 12, 13	fvmptd 6288	1 |- ( U e. (UnifOn ` X ) -> (toUnifSp ` U ) = ( { <. ( Base ` ndx ) , dom U. U >. , <. ( UnifSet ` ndx ) , U >. } sSet <. (TopSet ` ndx ) , (unifTop ` U ) >. ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   {cpr 4179   <.cop 4183   U.cuni 4436   |-> cmpt 4729   dom cdm 5114   ran crn 5115   ` cfv 5888  (class class class)co 6650   ndxcnx 15854   sSet csts 15855   Basecbs 15857  TopSetcts 15947   UnifSetcunif 15951  UnifOncust 22003  unifTopcutop 22034  toUnifSpctus 22059

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-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-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-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-iota 5851  df-fun 5890  df-fn 5891  df-fv 5896  df-ov 6653  df-ust 22004  df-tus 22062

This theorem is referenced by:  tuslem  22071