Theorem mexval 31399

Description: The set of expressions, which are pairs whose first element is a typecode, and whose second element is a raw expression. (Contributed by Mario Carneiro, 18-Jul-2016.)

Hypotheses

Ref	Expression
mexval.k	|- K = (mTC ` T )
mexval.e	|- E = (mEx ` T )
mexval.r	|- R = (mREx ` T )

Assertion

Ref	Expression
mexval	|- E = ( K X. R )

Proof of Theorem mexval

Dummy variable t is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mexval.e	. 2 |- E = (mEx ` T )
2		fveq2 6191	. . . . . 6 |- ( t = T -> (mTC ` t ) = (mTC ` T ) )
3		mexval.k	. . . . . 6 |- K = (mTC ` T )
4	2, 3	syl6eqr 2674	. . . . 5 |- ( t = T -> (mTC ` t ) = K )
5		fveq2 6191	. . . . . 6 |- ( t = T -> (mREx ` t ) = (mREx ` T ) )
6		mexval.r	. . . . . 6 |- R = (mREx ` T )
7	5, 6	syl6eqr 2674	. . . . 5 |- ( t = T -> (mREx ` t ) = R )
8	4, 7	xpeq12d 5140	. . . 4 |- ( t = T -> ( (mTC ` t ) X. (mREx ` t ) ) = ( K X. R ) )
9		df-mex 31384	. . . 4 |- mEx = ( t e. _V |-> ( (mTC ` t ) X. (mREx ` t ) ) )
10		fvex 6201	. . . . 5 |- (mTC ` t ) e. _V
11		fvex 6201	. . . . 5 |- (mREx ` t ) e. _V
12	10, 11	xpex 6962	. . . 4 |- ( (mTC ` t ) X. (mREx ` t ) ) e. _V
13	8, 9, 12	fvmpt3i 6287	. . 3 |- ( T e. _V -> (mEx ` T ) = ( K X. R ) )
14		xp0 5552	. . . . 5 |- ( K X. (/) ) = (/)
15	14	eqcomi 2631	. . . 4 |- (/) = ( K X. (/) )
16		fvprc 6185	. . . 4 |- ( -. T e. _V -> (mEx ` T ) = (/) )
17		fvprc 6185	. . . . . 6 |- ( -. T e. _V -> (mREx ` T ) = (/) )
18	6, 17	syl5eq 2668	. . . . 5 |- ( -. T e. _V -> R = (/) )
19	18	xpeq2d 5139	. . . 4 |- ( -. T e. _V -> ( K X. R ) = ( K X. (/) ) )
20	15, 16, 19	3eqtr4a 2682	. . 3 |- ( -. T e. _V -> (mEx ` T ) = ( K X. R ) )
21	13, 20	pm2.61i 176	. 2 |- (mEx ` T ) = ( K X. R )
22	1, 21	eqtri 2644	1 |- E = ( K X. R )

Syntax hints:   -. wn 3   = wceq 1483   e. wcel 1990   _Vcvv 3200   (/)c0 3915   X. cxp 5112   ` cfv 5888  mTCcmtc 31361  mRExcmrex 31363  mExcmex 31364

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-iota 5851  df-fun 5890  df-fv 5896  df-mex 31384

This theorem is referenced by:  mexval2  31400  msubff  31427  msubco  31428  msubff1  31453  mvhf  31455  msubvrs  31457