Theorem riinint 4993

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 4228	. . . . . . 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 2600	. . . . 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 4990	. . . 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 3408	. 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 3959	. . 3 |- |^| ( { X } u. ran ( k e. I |-> S ) ) = ( |^| { X } i^i |^| ran ( k e. I |-> S ) )
9		intsng 3962	. . . . 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 3407	. . 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 2276	. 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 2267	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 1397   e. wcel 2202   A.wral 2510   _Vcvv 2802   u. cun 3198   i^i cin 3199   C_ wss 3200   {csn 3669   |^|cint 3928   |^|_ciin 3971   |-> cmpt 4150   ran crn 4726

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-int 3929  df-iin 3973  df-br 4089  df-opab 4151  df-mpt 4152  df-cnv 4733  df-dm 4735  df-rn 4736

This theorem is referenced by: (None)