Metamath Proof Explorer


Theorem mvtinf

Description: Each variable typecode has infinitely many variables. (Contributed by Mario Carneiro, 18-Jul-2016)

Ref Expression
Hypotheses mvtinf.f F = mVT T
mvtinf.y Y = mType T
Assertion mvtinf T mFS X F ¬ Y -1 X Fin

Proof

Step Hyp Ref Expression
1 mvtinf.f F = mVT T
2 mvtinf.y Y = mType T
3 eqid mCN T = mCN T
4 eqid mVR T = mVR T
5 eqid mTC T = mTC T
6 eqid mAx T = mAx T
7 eqid mStat T = mStat T
8 3 4 2 1 5 6 7 ismfs T mFS T mFS mCN T mVR T = Y : mVR T mTC T mAx T mStat T v F ¬ Y -1 v Fin
9 8 ibi T mFS mCN T mVR T = Y : mVR T mTC T mAx T mStat T v F ¬ Y -1 v Fin
10 9 simprrd T mFS v F ¬ Y -1 v Fin
11 sneq v = X v = X
12 11 imaeq2d v = X Y -1 v = Y -1 X
13 12 eleq1d v = X Y -1 v Fin Y -1 X Fin
14 13 notbid v = X ¬ Y -1 v Fin ¬ Y -1 X Fin
15 14 rspccva v F ¬ Y -1 v Fin X F ¬ Y -1 X Fin
16 10 15 sylan T mFS X F ¬ Y -1 X Fin