Theorem mthmval 31472

Description: A theorem is a pre-statement, whose reduct is also the reduct of a provable pre-statement. Unlike the difference between pre-statement and statement, this application of the reduct is not necessarily trivial: there are theorems that are not themselves provable but are provable once enough "dummy variables" are introduced. (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
mthmval	|- U = ( `' R " ( R " J ) )

Proof of Theorem mthmval

Dummy variable t is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mthmval.u	. 2 |- U = (mThm ` T )
2		fveq2 6191	. . . . . . 7 |- ( t = T -> (mStRed ` t ) = (mStRed ` T ) )
3		mthmval.r	. . . . . . 7 |- R = (mStRed ` T )
4	2, 3	syl6eqr 2674	. . . . . 6 |- ( t = T -> (mStRed ` t ) = R )
5	4	cnveqd 5298	. . . . 5 |- ( t = T -> `' (mStRed ` t ) = `' R )
6		fveq2 6191	. . . . . . 7 |- ( t = T -> (mPPSt ` t ) = (mPPSt ` T ) )
7		mthmval.j	. . . . . . 7 |- J = (mPPSt ` T )
8	6, 7	syl6eqr 2674	. . . . . 6 |- ( t = T -> (mPPSt ` t ) = J )
9	4, 8	imaeq12d 5467	. . . . 5 |- ( t = T -> ( (mStRed ` t ) " (mPPSt ` t ) ) = ( R " J ) )
10	5, 9	imaeq12d 5467	. . . 4 |- ( t = T -> ( `' (mStRed ` t ) " ( (mStRed ` t ) " (mPPSt ` t ) ) ) = ( `' R " ( R " J ) ) )
11		df-mthm 31396	. . . 4 |- mThm = ( t e. _V |-> ( `' (mStRed ` t ) " ( (mStRed ` t ) " (mPPSt ` t ) ) ) )
12		fvex 6201	. . . . . 6 |- (mStRed ` t ) e. _V
13	12	cnvex 7113	. . . . 5 |- `' (mStRed ` t ) e. _V
14		imaexg 7103	. . . . 5 |- ( `' (mStRed ` t ) e. _V -> ( `' (mStRed ` t ) " ( (mStRed ` t ) " (mPPSt ` t ) ) ) e. _V )
15	13, 14	ax-mp 5	. . . 4 |- ( `' (mStRed ` t ) " ( (mStRed ` t ) " (mPPSt ` t ) ) ) e. _V
16	10, 11, 15	fvmpt3i 6287	. . 3 |- ( T e. _V -> (mThm ` T ) = ( `' R " ( R " J ) ) )
17		0ima 5482	. . . . 5 |- ( (/) " ( R " J ) ) = (/)
18	17	eqcomi 2631	. . . 4 |- (/) = ( (/) " ( R " J ) )
19		fvprc 6185	. . . 4 |- ( -. T e. _V -> (mThm ` T ) = (/) )
20		fvprc 6185	. . . . . . . 8 |- ( -. T e. _V -> (mStRed ` T ) = (/) )
21	3, 20	syl5eq 2668	. . . . . . 7 |- ( -. T e. _V -> R = (/) )
22	21	cnveqd 5298	. . . . . 6 |- ( -. T e. _V -> `' R = `' (/) )
23		cnv0 5535	. . . . . 6 |- `' (/) = (/)
24	22, 23	syl6eq 2672	. . . . 5 |- ( -. T e. _V -> `' R = (/) )
25	24	imaeq1d 5465	. . . 4 |- ( -. T e. _V -> ( `' R " ( R " J ) ) = ( (/) " ( R " J ) ) )
26	18, 19, 25	3eqtr4a 2682	. . 3 |- ( -. T e. _V -> (mThm ` T ) = ( `' R " ( R " J ) ) )
27	16, 26	pm2.61i 176	. 2 |- (mThm ` T ) = ( `' R " ( R " J ) )
28	1, 27	eqtri 2644	1 |- U = ( `' R " ( R " J ) )

Syntax hints:   -. wn 3   = wceq 1483   e. wcel 1990   _Vcvv 3200   (/)c0 3915   `'ccnv 5113   "cima 5117   ` cfv 5888  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-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-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fv 5896  df-mthm 31396

This theorem is referenced by:  elmthm  31473  mthmsta  31475  mthmblem  31477