Theorem riinint 4611

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 3917	. . . . . . 7 |- ( ( S C_ X /\ X e. V ) -> S e. _V )
2	1	expcom 114	. . . . . 6 |- ( X e. V -> ( S C_ X -> S e. _V ) )
3	2	ralimdv 2430	. . . . 5 |- ( X e. V -> ( A. k e. I S C_ X -> A. k e. I S e. _V ) )
4	3	imp 122	. . . 4 |- ( ( X e. V /\ A. k e. I S C_ X ) -> A. k e. I S e. _V )
5		dfiin3g 4608	. . . 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 3167	. 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 3667	. . 3 |- |^| ( { X } u. ran ( k e. I |-> S ) ) = ( |^| { X } i^i |^| ran ( k e. I |-> S ) )
9		intsng 3670	. . . . 5 |- ( X e. V -> |^| { X } = X )
10	9	adantr 270	. . . 4 |- ( ( X e. V /\ A. k e. I S C_ X ) -> |^| { X } = X )
11	10	ineq1d 3166	. . 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 2125	. 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 2116	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 102   = wceq 1284   e. wcel 1433   A.wral 2348   _Vcvv 2601   u. cun 2971   i^i cin 2972   C_ wss 2973   {csn 3398   |^|cint 3636   |^|_ciin 3679   |-> cmpt 3839   ran crn 4364

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-int 3637  df-iin 3681  df-br 3786  df-opab 3840  df-mpt 3841  df-cnv 4371  df-dm 4373  df-rn 4374

This theorem is referenced by: (None)