Theorem msrrcl 31440

Description: If X and Y have the same reduct, then one is a pre-statement iff the other is. (Contributed by Mario Carneiro, 18-Jul-2016.)

Hypotheses

Ref	Expression
mpstssv.p	|- P = (mPreSt ` T )
msrf.r	|- R = (mStRed ` T )

Assertion

Ref	Expression
msrrcl	|- ( ( R ` X ) = ( R ` Y ) -> ( X e. P <-> Y e. P ) )

Proof of Theorem msrrcl

Step	Hyp	Ref	Expression
1		mpstssv.p	. . . . 5 |- P = (mPreSt ` T )
2		msrf.r	. . . . 5 |- R = (mStRed ` T )
3	1, 2	msrf 31439	. . . 4 |- R : P --> P
4	3	ffvelrni 6358	. . 3 |- ( X e. P -> ( R ` X ) e. P )
5	4	a1i 11	. 2 |- ( ( R ` X ) = ( R ` Y ) -> ( X e. P -> ( R ` X ) e. P ) )
6	3	ffvelrni 6358	. . 3 |- ( Y e. P -> ( R ` Y ) e. P )
7		eleq1 2689	. . 3 |- ( ( R ` X ) = ( R ` Y ) -> ( ( R ` X ) e. P <-> ( R ` Y ) e. P ) )
8	6, 7	syl5ibr 236	. 2 |- ( ( R ` X ) = ( R ` Y ) -> ( Y e. P -> ( R ` X ) e. P ) )
9	3	fdmi 6052	. . . . . 6 |- dom R = P
10		0nelxp 5143	. . . . . . 7 |- -. (/) e. ( ( _V X. _V ) X. _V )
11	1	mpstssv 31436	. . . . . . . 8 |- P C_ ( ( _V X. _V ) X. _V )
12	11	sseli 3599	. . . . . . 7 |- ( (/) e. P -> (/) e. ( ( _V X. _V ) X. _V ) )
13	10, 12	mto 188	. . . . . 6 |- -. (/) e. P
14	9, 13	ndmfvrcl 6219	. . . . 5 |- ( ( R ` X ) e. P -> X e. P )
15	14	adantl 482	. . . 4 |- ( ( ( R ` X ) = ( R ` Y ) /\ ( R ` X ) e. P ) -> X e. P )
16	7	biimpa 501	. . . . 5 |- ( ( ( R ` X ) = ( R ` Y ) /\ ( R ` X ) e. P ) -> ( R ` Y ) e. P )
17	9, 13	ndmfvrcl 6219	. . . . 5 |- ( ( R ` Y ) e. P -> Y e. P )
18	16, 17	syl 17	. . . 4 |- ( ( ( R ` X ) = ( R ` Y ) /\ ( R ` X ) e. P ) -> Y e. P )
19	15, 18	2thd 255	. . 3 |- ( ( ( R ` X ) = ( R ` Y ) /\ ( R ` X ) e. P ) -> ( X e. P <-> Y e. P ) )
20	19	ex 450	. 2 |- ( ( R ` X ) = ( R ` Y ) -> ( ( R ` X ) e. P -> ( X e. P <-> Y e. P ) ) )
21	5, 8, 20	pm5.21ndd 369	1 |- ( ( R ` X ) = ( R ` Y ) -> ( X e. P <-> Y e. P ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   (/)c0 3915   X. cxp 5112   ` cfv 5888  mPreStcmpst 31370  mStRedcmsr 31371

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-fal 1489  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-ot 4186  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-1st 7168  df-2nd 7169  df-mpst 31390  df-msr 31391

This theorem is referenced by:  elmthm  31473