Theorem mthmsta 31475

Description: A theorem is a pre-statement. (Contributed by Mario Carneiro, 18-Jul-2016.)

Hypotheses

Ref	Expression
mthmsta.u	|- U = (mThm ` T )
mthmsta.s	|- S = (mPreSt ` T )

Assertion

Ref	Expression
mthmsta	|- U C_ S

Proof of Theorem mthmsta

Step	Hyp	Ref	Expression
1		eqid 2622	. . 3 |- (mStRed ` T ) = (mStRed ` T )
2		eqid 2622	. . 3 |- (mPPSt ` T ) = (mPPSt ` T )
3		mthmsta.u	. . 3 |- U = (mThm ` T )
4	1, 2, 3	mthmval 31472	. 2 |- U = ( `' (mStRed ` T ) " ( (mStRed ` T ) " (mPPSt ` T ) ) )
5		cnvimass 5485	. . 3 |- ( `' (mStRed ` T ) " ( (mStRed ` T ) " (mPPSt ` T ) ) ) C_ dom (mStRed ` T )
6		mthmsta.s	. . . . 5 |- S = (mPreSt ` T )
7	6, 1	msrf 31439	. . . 4 |- (mStRed ` T ) : S --> S
8	7	fdmi 6052	. . 3 |- dom (mStRed ` T ) = S
9	5, 8	sseqtri 3637	. 2 |- ( `' (mStRed ` T ) " ( (mStRed ` T ) " (mPPSt ` T ) ) ) C_ S
10	4, 9	eqsstri 3635	1 |- U C_ S

Syntax hints:   = wceq 1483   C_ wss 3574   `'ccnv 5113   dom cdm 5114   "cima 5117   ` 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-1st 7168  df-2nd 7169  df-mpst 31390  df-msr 31391  df-mthm 31396

This theorem is referenced by:  mthmpps  31479