Theorem dfiin3g 4990

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 4003	. 2 |- ( A. x e. A B e. C -> |^|_ x e. A B = |^| { y | E. x e. A y = B } )
2		eqid 2231	. . . 4 |- ( x e. A |-> B ) = ( x e. A |-> B )
3	2	rnmpt 4980	. . 3 |- ran ( x e. A |-> B ) = { y | E. x e. A y = B }
4	3	inteqi 3932	. 2 |- |^| ran ( x e. A |-> B ) = |^| { y | E. x e. A y = B }
5	1, 4	eqtr4di 2282	1 |- ( A. x e. A B e. C -> |^|_ x e. A B = |^| ran ( x e. A |-> B ) )

Syntax hints:   -> wi 4   = wceq 1397   e. wcel 2202   {cab 2217   A.wral 2510   E.wrex 2511   |^|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:  dfiin3  4992  riinint  4993