Theorem elmthm 31473

Description: A theorem is a pre-statement, whose reduct is also the reduct of a provable pre-statement. (Contributed by Mario Carneiro, 18-Jul-2016.)

Hypotheses

Ref	Expression
mthmval.r	|- R = (mStRed ` T )
mthmval.j	|- J = (mPPSt ` T )
mthmval.u	|- U = (mThm ` T )

Assertion

Ref	Expression
elmthm	|- ( X e. U <-> E. x e. J ( R ` x ) = ( R ` X ) )

Distinct variable groups:   x, J   x, R   x, T   x, X

Allowed substitution hint:   U( x)

Proof of Theorem elmthm

Step	Hyp	Ref	Expression
1		mthmval.r	. . . 4 |- R = (mStRed ` T )
2		mthmval.j	. . . 4 |- J = (mPPSt ` T )
3		mthmval.u	. . . 4 |- U = (mThm ` T )
4	1, 2, 3	mthmval 31472	. . 3 |- U = ( `' R " ( R " J ) )
5	4	eleq2i 2693	. 2 |- ( X e. U <-> X e. ( `' R " ( R " J ) ) )
6		eqid 2622	. . . . 5 |- (mPreSt ` T ) = (mPreSt ` T )
7	6, 1	msrf 31439	. . . 4 |- R : (mPreSt ` T ) --> (mPreSt ` T )
8		ffn 6045	. . . 4 |- ( R : (mPreSt ` T ) --> (mPreSt ` T ) -> R Fn (mPreSt ` T ) )
9	7, 8	ax-mp 5	. . 3 |- R Fn (mPreSt ` T )
10		elpreima 6337	. . 3 |- ( R Fn (mPreSt ` T ) -> ( X e. ( `' R " ( R " J ) ) <-> ( X e. (mPreSt ` T ) /\ ( R ` X ) e. ( R " J ) ) ) )
11	9, 10	ax-mp 5	. 2 |- ( X e. ( `' R " ( R " J ) ) <-> ( X e. (mPreSt ` T ) /\ ( R ` X ) e. ( R " J ) ) )
12	6, 2	mppspst 31471	. . . . 5 |- J C_ (mPreSt ` T )
13		fvelimab 6253	. . . . 5 |- ( ( R Fn (mPreSt ` T ) /\ J C_ (mPreSt ` T ) ) -> ( ( R ` X ) e. ( R " J ) <-> E. x e. J ( R ` x ) = ( R ` X ) ) )
14	9, 12, 13	mp2an 708	. . . 4 |- ( ( R ` X ) e. ( R " J ) <-> E. x e. J ( R ` x ) = ( R ` X ) )
15	14	anbi2i 730	. . 3 |- ( ( X e. (mPreSt ` T ) /\ ( R ` X ) e. ( R " J ) ) <-> ( X e. (mPreSt ` T ) /\ E. x e. J ( R ` x ) = ( R ` X ) ) )
16	12	sseli 3599	. . . . . 6 |- ( x e. J -> x e. (mPreSt ` T ) )
17	6, 1	msrrcl 31440	. . . . . 6 |- ( ( R ` x ) = ( R ` X ) -> ( x e. (mPreSt ` T ) <-> X e. (mPreSt ` T ) ) )
18	16, 17	syl5ibcom 235	. . . . 5 |- ( x e. J -> ( ( R ` x ) = ( R ` X ) -> X e. (mPreSt ` T ) ) )
19	18	rexlimiv 3027	. . . 4 |- ( E. x e. J ( R ` x ) = ( R ` X ) -> X e. (mPreSt ` T ) )
20	19	pm4.71ri 665	. . 3 |- ( E. x e. J ( R ` x ) = ( R ` X ) <-> ( X e. (mPreSt ` T ) /\ E. x e. J ( R ` x ) = ( R ` X ) ) )
21	15, 20	bitr4i 267	. 2 |- ( ( X e. (mPreSt ` T ) /\ ( R ` X ) e. ( R " J ) ) <-> E. x e. J ( R ` x ) = ( R ` X ) )
22	5, 11, 21	3bitri 286	1 |- ( X e. U <-> E. x e. J ( R ` x ) = ( R ` X ) )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   C_ wss 3574   `'ccnv 5113   "cima 5117   Fn wfn 5883   -->wf 5884   ` cfv 5888  mPreStcmpst 31370  mStRedcmsr 31371  mPPStcmpps 31375  mThmcmthm 31376

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-ov 6653  df-oprab 6654  df-1st 7168  df-2nd 7169  df-mpst 31390  df-msr 31391  df-mpps 31395  df-mthm 31396

This theorem is referenced by:  mthmi  31474  mthmpps  31479