Theorem dfiin3g 5015

Description: Alternate definition of indexed intersection when B is a set. (Contributed by Mario Carneiro, 31-Aug-2015.)

Assertion

Ref	Expression
dfiin3g	|- ( A. x e. A B e. C -> |^|_ x e. A B = |^| ran ( x e. A |-> B ) )

Proof of Theorem dfiin3g

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfiin2g 4024	. 2 |- ( A. x e. A B e. C -> |^|_ x e. A B = |^| { y | E. x e. A y = B } )
2		eqid 2232	. . . 4 |- ( x e. A |-> B ) = ( x e. A |-> B )
3	2	rnmpt 5005	. . 3 |- ran ( x e. A |-> B ) = { y | E. x e. A y = B }
4	3	inteqi 3953	. 2 |- |^| ran ( x e. A |-> B ) = |^| { y | E. x e. A y = B }
5	1, 4	eqtr4di 2283	1 |- ( A. x e. A B e. C -> |^|_ x e. A B = |^| ran ( x e. A |-> B ) )

Syntax hints:   -> wi 4   = wceq 1398   e. wcel 2203   {cab 2218   A.wral 2520   E.wrex 2521   |^|cint 3949   |^|_ciin 3992   |-> cmpt 4171   ran crn 4750

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 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2815  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-int 3950  df-iin 3994  df-br 4110  df-opab 4172  df-mpt 4173  df-cnv 4757  df-dm 4759  df-rn 4760

This theorem is referenced by:  dfiin3  5017  riinint  5018