Theorem riinint 4999

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 4233	. . . . . . 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 2601	. . . . 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 4996	. . . 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 3410	. 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 3964	. . 3 |- |^| ( { X } u. ran ( k e. I |-> S ) ) = ( |^| { X } i^i |^| ran ( k e. I |-> S ) )
9		intsng 3967	. . . . 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 3409	. . 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 1398   e. wcel 2202   A.wral 2511   _Vcvv 2803   u. cun 3199   i^i cin 3200   C_ wss 3201   {csn 3673   |^|cint 3933   |^|_ciin 3976   |-> cmpt 4155   ran crn 4732

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-int 3934  df-iin 3978  df-br 4094  df-opab 4156  df-mpt 4157  df-cnv 4739  df-dm 4741  df-rn 4742

This theorem is referenced by: (None)