Theorem elmpst 31433

Description: Property of being a pre-statement. (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
elmpst	|- ( <. D , H , A >. e. P <-> ( ( D C_ V /\ `' D = D ) /\ ( H C_ E /\ H e. Fin ) /\ A e. E ) )

Proof of Theorem elmpst

Dummy variable d is distinct from all other variables.

Step	Hyp	Ref	Expression
1		opelxp 5146	. . 3 |- ( <. <. D , H >. , A >. e. ( ( { d e. ~P V | `' d = d } X. ( ~P E i^i Fin ) ) X. E ) <-> ( <. D , H >. e. ( { d e. ~P V | `' d = d } X. ( ~P E i^i Fin ) ) /\ A e. E ) )
2		opelxp 5146	. . . . 5 |- ( <. D , H >. e. ( { d e. ~P V | `' d = d } X. ( ~P E i^i Fin ) ) <-> ( D e. { d e. ~P V | `' d = d } /\ H e. ( ~P E i^i Fin ) ) )
3		cnveq 5296	. . . . . . . . 9 |- ( d = D -> `' d = `' D )
4		id 22	. . . . . . . . 9 |- ( d = D -> d = D )
5	3, 4	eqeq12d 2637	. . . . . . . 8 |- ( d = D -> ( `' d = d <-> `' D = D ) )
6	5	elrab 3363	. . . . . . 7 |- ( D e. { d e. ~P V | `' d = d } <-> ( D e. ~P V /\ `' D = D ) )
7		mpstval.v	. . . . . . . . . 10 |- V = (mDV ` T )
8		fvex 6201	. . . . . . . . . 10 |- (mDV ` T ) e. _V
9	7, 8	eqeltri 2697	. . . . . . . . 9 |- V e. _V
10	9	elpw2 4828	. . . . . . . 8 |- ( D e. ~P V <-> D C_ V )
11	10	anbi1i 731	. . . . . . 7 |- ( ( D e. ~P V /\ `' D = D ) <-> ( D C_ V /\ `' D = D ) )
12	6, 11	bitri 264	. . . . . 6 |- ( D e. { d e. ~P V | `' d = d } <-> ( D C_ V /\ `' D = D ) )
13		elfpw 8268	. . . . . 6 |- ( H e. ( ~P E i^i Fin ) <-> ( H C_ E /\ H e. Fin ) )
14	12, 13	anbi12i 733	. . . . 5 |- ( ( D e. { d e. ~P V | `' d = d } /\ H e. ( ~P E i^i Fin ) ) <-> ( ( D C_ V /\ `' D = D ) /\ ( H C_ E /\ H e. Fin ) ) )
15	2, 14	bitri 264	. . . 4 |- ( <. D , H >. e. ( { d e. ~P V | `' d = d } X. ( ~P E i^i Fin ) ) <-> ( ( D C_ V /\ `' D = D ) /\ ( H C_ E /\ H e. Fin ) ) )
16	15	anbi1i 731	. . 3 |- ( ( <. D , H >. e. ( { d e. ~P V | `' d = d } X. ( ~P E i^i Fin ) ) /\ A e. E ) <-> ( ( ( D C_ V /\ `' D = D ) /\ ( H C_ E /\ H e. Fin ) ) /\ A e. E ) )
17	1, 16	bitri 264	. 2 |- ( <. <. D , H >. , A >. e. ( ( { d e. ~P V | `' d = d } X. ( ~P E i^i Fin ) ) X. E ) <-> ( ( ( D C_ V /\ `' D = D ) /\ ( H C_ E /\ H e. Fin ) ) /\ A e. E ) )
18		df-ot 4186	. . 3 |- <. D , H , A >. = <. <. D , H >. , A >.
19		mpstval.e	. . . 4 |- E = (mEx ` T )
20		mpstval.p	. . . 4 |- P = (mPreSt ` T )
21	7, 19, 20	mpstval 31432	. . 3 |- P = ( ( { d e. ~P V | `' d = d } X. ( ~P E i^i Fin ) ) X. E )
22	18, 21	eleq12i 2694	. 2 |- ( <. D , H , A >. e. P <-> <. <. D , H >. , A >. e. ( ( { d e. ~P V | `' d = d } X. ( ~P E i^i Fin ) ) X. E ) )
23		df-3an 1039	. 2 |- ( ( ( D C_ V /\ `' D = D ) /\ ( H C_ E /\ H e. Fin ) /\ A e. E ) <-> ( ( ( D C_ V /\ `' D = D ) /\ ( H C_ E /\ H e. Fin ) ) /\ A e. E ) )
24	17, 22, 23	3bitr4i 292	1 |- ( <. D , H , A >. e. P <-> ( ( D C_ V /\ `' D = D ) /\ ( H C_ E /\ H e. Fin ) /\ A e. E ) )

Syntax hints:   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   {crab 2916   _Vcvv 3200   i^i cin 3573   C_ wss 3574   ~Pcpw 4158   <.cop 4183   <.cotp 4185   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-ot 4186  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:  msrval  31435  msrf  31439  mclsssvlem  31459  mclsax  31466  mclsind  31467  mthmpps  31479  mclsppslem  31480  mclspps  31481