Theorem mpstval 31432

Description: A pre-statement is an ordered triple, whose first member is a symmetric set of dv conditions, whose second member is a finite set of expressions, and whose third member is an expression. (Contributed by Mario Carneiro, 18-Jul-2016.)

Hypotheses

Ref	Expression
mpstval.v	|- V = (mDV ` T )
mpstval.e	|- E = (mEx ` T )
mpstval.p	|- P = (mPreSt ` T )

Assertion

Ref	Expression
mpstval	|- P = ( ( { d e. ~P V | `' d = d } X. ( ~P E i^i Fin ) ) X. E )

Distinct variable groups:   T, d   V, d

Allowed substitution hints:   P( d)   E( d)

Proof of Theorem mpstval

Dummy variable t is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mpstval.p	. 2 |- P = (mPreSt ` T )
2		fveq2 6191	. . . . . . . . 9 |- ( t = T -> (mDV ` t ) = (mDV ` T ) )
3		mpstval.v	. . . . . . . . 9 |- V = (mDV ` T )
4	2, 3	syl6eqr 2674	. . . . . . . 8 |- ( t = T -> (mDV ` t ) = V )
5	4	pweqd 4163	. . . . . . 7 |- ( t = T -> ~P (mDV ` t ) = ~P V )
6	5	rabeqdv 3194	. . . . . 6 |- ( t = T -> { d e. ~P (mDV ` t ) | `' d = d } = { d e. ~P V | `' d = d } )
7		fveq2 6191	. . . . . . . . 9 |- ( t = T -> (mEx ` t ) = (mEx ` T ) )
8		mpstval.e	. . . . . . . . 9 |- E = (mEx ` T )
9	7, 8	syl6eqr 2674	. . . . . . . 8 |- ( t = T -> (mEx ` t ) = E )
10	9	pweqd 4163	. . . . . . 7 |- ( t = T -> ~P (mEx ` t ) = ~P E )
11	10	ineq1d 3813	. . . . . 6 |- ( t = T -> ( ~P (mEx ` t ) i^i Fin ) = ( ~P E i^i Fin ) )
12	6, 11	xpeq12d 5140	. . . . 5 |- ( t = T -> ( { d e. ~P (mDV ` t ) | `' d = d } X. ( ~P (mEx ` t ) i^i Fin ) ) = ( { d e. ~P V | `' d = d } X. ( ~P E i^i Fin ) ) )
13	12, 9	xpeq12d 5140	. . . 4 |- ( t = T -> ( ( { d e. ~P (mDV ` t ) | `' d = d } X. ( ~P (mEx ` t ) i^i Fin ) ) X. (mEx ` t ) ) = ( ( { d e. ~P V | `' d = d } X. ( ~P E i^i Fin ) ) X. E ) )
14		df-mpst 31390	. . . 4 |- mPreSt = ( t e. _V |-> ( ( { d e. ~P (mDV ` t ) | `' d = d } X. ( ~P (mEx ` t ) i^i Fin ) ) X. (mEx ` t ) ) )
15		fvex 6201	. . . . . . . . 9 |- (mDV ` T ) e. _V
16	3, 15	eqeltri 2697	. . . . . . . 8 |- V e. _V
17	16	pwex 4848	. . . . . . 7 |- ~P V e. _V
18	17	rabex 4813	. . . . . 6 |- { d e. ~P V | `' d = d } e. _V
19		fvex 6201	. . . . . . . . 9 |- (mEx ` T ) e. _V
20	8, 19	eqeltri 2697	. . . . . . . 8 |- E e. _V
21	20	pwex 4848	. . . . . . 7 |- ~P E e. _V
22	21	inex1 4799	. . . . . 6 |- ( ~P E i^i Fin ) e. _V
23	18, 22	xpex 6962	. . . . 5 |- ( { d e. ~P V | `' d = d } X. ( ~P E i^i Fin ) ) e. _V
24	23, 20	xpex 6962	. . . 4 |- ( ( { d e. ~P V | `' d = d } X. ( ~P E i^i Fin ) ) X. E ) e. _V
25	13, 14, 24	fvmpt 6282	. . 3 |- ( T e. _V -> (mPreSt ` T ) = ( ( { d e. ~P V | `' d = d } X. ( ~P E i^i Fin ) ) X. E ) )
26		xp0 5552	. . . . 5 |- ( ( { d e. ~P V | `' d = d } X. ( ~P E i^i Fin ) ) X. (/) ) = (/)
27	26	eqcomi 2631	. . . 4 |- (/) = ( ( { d e. ~P V | `' d = d } X. ( ~P E i^i Fin ) ) X. (/) )
28		fvprc 6185	. . . 4 |- ( -. T e. _V -> (mPreSt ` T ) = (/) )
29		fvprc 6185	. . . . . 6 |- ( -. T e. _V -> (mEx ` T ) = (/) )
30	8, 29	syl5eq 2668	. . . . 5 |- ( -. T e. _V -> E = (/) )
31	30	xpeq2d 5139	. . . 4 |- ( -. T e. _V -> ( ( { d e. ~P V | `' d = d } X. ( ~P E i^i Fin ) ) X. E ) = ( ( { d e. ~P V | `' d = d } X. ( ~P E i^i Fin ) ) X. (/) ) )
32	27, 28, 31	3eqtr4a 2682	. . 3 |- ( -. T e. _V -> (mPreSt ` T ) = ( ( { d e. ~P V | `' d = d } X. ( ~P E i^i Fin ) ) X. E ) )
33	25, 32	pm2.61i 176	. 2 |- (mPreSt ` T ) = ( ( { d e. ~P V | `' d = d } X. ( ~P E i^i Fin ) ) X. E )
34	1, 33	eqtri 2644	1 |- P = ( ( { d e. ~P V | `' d = d } X. ( ~P E i^i Fin ) ) X. E )

Syntax hints:   -. wn 3   = wceq 1483   e. wcel 1990   {crab 2916   _Vcvv 3200   i^i cin 3573   (/)c0 3915   ~Pcpw 4158   X. cxp 5112   `'ccnv 5113   ` cfv 5888   Fincfn 7955  mExcmex 31364  mDVcmdv 31365  mPreStcmpst 31370

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-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  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-iota 5851  df-fun 5890  df-fv 5896  df-mpst 31390

This theorem is referenced by:  elmpst  31433  mpstssv  31436