Theorem riinint 4865

Description: Express a relative indexed intersection as an intersection. (Contributed by Stefan O'Rear, 22-Feb-2015.)

Assertion

Ref	Expression
riinint	|- ( ( X e. V /\ A. k e. I S C_ X ) -> ( X i^i |^|_ k e. I S ) = |^| ( { X } u. ran ( k e. I |-> S ) ) )

Distinct variable groups:   k, V   k, X

Allowed substitution hints:   S( k)   I( k)

Proof of Theorem riinint

Step	Hyp	Ref	Expression
1		ssexg 4121	. . . . . . 7 |- ( ( S C_ X /\ X e. V ) -> S e. _V )
2	1	expcom 115	. . . . . 6 |- ( X e. V -> ( S C_ X -> S e. _V ) )
3	2	ralimdv 2534	. . . . 5 |- ( X e. V -> ( A. k e. I S C_ X -> A. k e. I S e. _V ) )
4	3	imp 123	. . . 4 |- ( ( X e. V /\ A. k e. I S C_ X ) -> A. k e. I S e. _V )
5		dfiin3g 4862	. . . 4 |- ( A. k e. I S e. _V -> |^|_ k e. I S = |^| ran ( k e. I |-> S ) )
6	4, 5	syl 14	. . 3 |- ( ( X e. V /\ A. k e. I S C_ X ) -> |^|_ k e. I S = |^| ran ( k e. I |-> S ) )
7	6	ineq2d 3323	. 2 |- ( ( X e. V /\ A. k e. I S C_ X ) -> ( X i^i |^|_ k e. I S ) = ( X i^i |^| ran ( k e. I |-> S ) ) )
8		intun 3855	. . 3 |- |^| ( { X } u. ran ( k e. I |-> S ) ) = ( |^| { X } i^i |^| ran ( k e. I |-> S ) )
9		intsng 3858	. . . . 5 |- ( X e. V -> |^| { X } = X )
10	9	adantr 274	. . . 4 |- ( ( X e. V /\ A. k e. I S C_ X ) -> |^| { X } = X )
11	10	ineq1d 3322	. . 3 |- ( ( X e. V /\ A. k e. I S C_ X ) -> ( |^| { X } i^i |^| ran ( k e. I |-> S ) ) = ( X i^i |^| ran ( k e. I |-> S ) ) )
12	8, 11	syl5eq 2211	. 2 |- ( ( X e. V /\ A. k e. I S C_ X ) -> |^| ( { X } u. ran ( k e. I |-> S ) ) = ( X i^i |^| ran ( k e. I |-> S ) ) )
13	7, 12	eqtr4d 2201	1 |- ( ( X e. V /\ A. k e. I S C_ X ) -> ( X i^i |^|_ k e. I S ) = |^| ( { X } u. ran ( k e. I |-> S ) ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1343   e. wcel 2136   A.wral 2444   _Vcvv 2726   u. cun 3114   i^i cin 3115   C_ wss 3116   {csn 3576   |^|cint 3824   |^|_ciin 3867   |-> cmpt 4043   ran crn 4605

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-int 3825  df-iin 3869  df-br 3983  df-opab 4044  df-mpt 4045  df-cnv 4612  df-dm 4614  df-rn 4615

This theorem is referenced by: (None)