Theorem mdvval 31401

Description: The set of disjoint variable conditions, which are pairs of distinct variables. (This definition differs from appendix C, which uses unordered pairs instead. We use ordered pairs, but all sets of dv conditions of interest will be symmetric, so it does not matter.) (Contributed by Mario Carneiro, 18-Jul-2016.)

Hypotheses

Ref	Expression
mdvval.v	|- V = (mVR ` T )
mdvval.d	|- D = (mDV ` T )

Assertion

Ref	Expression
mdvval	|- D = ( ( V X. V ) \ _I )

Proof of Theorem mdvval

Dummy variable t is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mdvval.d	. 2 |- D = (mDV ` T )
2		fveq2 6191	. . . . . . 7 |- ( t = T -> (mVR ` t ) = (mVR ` T ) )
3		mdvval.v	. . . . . . 7 |- V = (mVR ` T )
4	2, 3	syl6eqr 2674	. . . . . 6 |- ( t = T -> (mVR ` t ) = V )
5	4	sqxpeqd 5141	. . . . 5 |- ( t = T -> ( (mVR ` t ) X. (mVR ` t ) ) = ( V X. V ) )
6	5	difeq1d 3727	. . . 4 |- ( t = T -> ( ( (mVR ` t ) X. (mVR ` t ) ) \ _I ) = ( ( V X. V ) \ _I ) )
7		df-mdv 31385	. . . 4 |- mDV = ( t e. _V |-> ( ( (mVR ` t ) X. (mVR ` t ) ) \ _I ) )
8		fvex 6201	. . . . . 6 |- (mVR ` t ) e. _V
9	8, 8	xpex 6962	. . . . 5 |- ( (mVR ` t ) X. (mVR ` t ) ) e. _V
10		difexg 4808	. . . . 5 |- ( ( (mVR ` t ) X. (mVR ` t ) ) e. _V -> ( ( (mVR ` t ) X. (mVR ` t ) ) \ _I ) e. _V )
11	9, 10	ax-mp 5	. . . 4 |- ( ( (mVR ` t ) X. (mVR ` t ) ) \ _I ) e. _V
12	6, 7, 11	fvmpt3i 6287	. . 3 |- ( T e. _V -> (mDV ` T ) = ( ( V X. V ) \ _I ) )
13		0dif 3977	. . . . 5 |- ( (/) \ _I ) = (/)
14	13	eqcomi 2631	. . . 4 |- (/) = ( (/) \ _I )
15		fvprc 6185	. . . 4 |- ( -. T e. _V -> (mDV ` T ) = (/) )
16		fvprc 6185	. . . . . . . 8 |- ( -. T e. _V -> (mVR ` T ) = (/) )
17	3, 16	syl5eq 2668	. . . . . . 7 |- ( -. T e. _V -> V = (/) )
18	17	xpeq2d 5139	. . . . . 6 |- ( -. T e. _V -> ( V X. V ) = ( V X. (/) ) )
19		xp0 5552	. . . . . 6 |- ( V X. (/) ) = (/)
20	18, 19	syl6eq 2672	. . . . 5 |- ( -. T e. _V -> ( V X. V ) = (/) )
21	20	difeq1d 3727	. . . 4 |- ( -. T e. _V -> ( ( V X. V ) \ _I ) = ( (/) \ _I ) )
22	14, 15, 21	3eqtr4a 2682	. . 3 |- ( -. T e. _V -> (mDV ` T ) = ( ( V X. V ) \ _I ) )
23	12, 22	pm2.61i 176	. 2 |- (mDV ` T ) = ( ( V X. V ) \ _I )
24	1, 23	eqtri 2644	1 |- D = ( ( V X. V ) \ _I )

Syntax hints:   -. wn 3   = wceq 1483   e. wcel 1990   _Vcvv 3200   \ cdif 3571   (/)c0 3915   _I cid 5023   X. cxp 5112   ` cfv 5888  mVRcmvar 31358  mDVcmdv 31365

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-mdv 31385

This theorem is referenced by:  mthmpps  31479  mclsppslem  31480