Theorem riinint 4924

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 4169	. . . . . . 7 |- ( ( S C_ X /\ X e. V ) -> S e. _V )
2	1	expcom 116	. . . . . 6 |- ( X e. V -> ( S C_ X -> S e. _V ) )
3	2	ralimdv 2562	. . . . 5 |- ( X e. V -> ( A. k e. I S C_ X -> A. k e. I S e. _V ) )
4	3	imp 124	. . . 4 |- ( ( X e. V /\ A. k e. I S C_ X ) -> A. k e. I S e. _V )
5		dfiin3g 4921	. . . 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 3361	. 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 3902	. . 3 |- |^| ( { X } u. ran ( k e. I |-> S ) ) = ( |^| { X } i^i |^| ran ( k e. I |-> S ) )
9		intsng 3905	. . . . 5 |- ( X e. V -> |^| { X } = X )
10	9	adantr 276	. . . 4 |- ( ( X e. V /\ A. k e. I S C_ X ) -> |^| { X } = X )
11	10	ineq1d 3360	. . 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	eqtrid 2238	. 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 2229	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 104   = wceq 1364   e. wcel 2164   A.wral 2472   _Vcvv 2760   u. cun 3152   i^i cin 3153   C_ wss 3154   {csn 3619   |^|cint 3871   |^|_ciin 3914   |-> cmpt 4091   ran crn 4661

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-int 3872  df-iin 3916  df-br 4031  df-opab 4092  df-mpt 4093  df-cnv 4668  df-dm 4670  df-rn 4671

This theorem is referenced by: (None)