Theorem riin0 4594

Description: Relative intersection of an empty family. (Contributed by Stefan O'Rear, 3-Apr-2015.)

Assertion

Ref	Expression
riin0	|- ( X = (/) -> ( A i^i |^|_ x e. X S ) = A )

Distinct variable groups:   x, A   x, X

Allowed substitution hint:   S( x)

Proof of Theorem riin0

Step	Hyp	Ref	Expression
1		iineq1 4535	. . 3 |- ( X = (/) -> |^|_ x e. X S = |^|_ x e. (/) S )
2	1	ineq2d 3814	. 2 |- ( X = (/) -> ( A i^i |^|_ x e. X S ) = ( A i^i |^|_ x e. (/) S ) )
3		0iin 4578	. . . 4 |- |^|_ x e. (/) S = _V
4	3	ineq2i 3811	. . 3 |- ( A i^i |^|_ x e. (/) S ) = ( A i^i _V )
5		inv1 3970	. . 3 |- ( A i^i _V ) = A
6	4, 5	eqtri 2644	. 2 |- ( A i^i |^|_ x e. (/) S ) = A
7	2, 6	syl6eq 2672	1 |- ( X = (/) -> ( A i^i |^|_ x e. X S ) = A )

Syntax hints:   -> wi 4   = wceq 1483   _Vcvv 3200   i^i cin 3573   (/)c0 3915   |^|_ciin 4521

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-v 3202  df-dif 3577  df-in 3581  df-ss 3588  df-nul 3916  df-iin 4523

This theorem is referenced by:  riinrab  4596  riiner  7820  mreriincl  16258  riinopn  20713  riincld  20848  fnemeet2  32362  pmapglb2N  35057  pmapglb2xN  35058