Theorem dfiin3g 4883

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 3919	. 2 |- ( A. x e. A B e. C -> |^|_ x e. A B = |^| { y | E. x e. A y = B } )
2		eqid 2177	. . . 4 |- ( x e. A |-> B ) = ( x e. A |-> B )
3	2	rnmpt 4873	. . 3 |- ran ( x e. A |-> B ) = { y | E. x e. A y = B }
4	3	inteqi 3848	. 2 |- |^| ran ( x e. A |-> B ) = |^| { y | E. x e. A y = B }
5	1, 4	eqtr4di 2228	1 |- ( A. x e. A B e. C -> |^|_ x e. A B = |^| ran ( x e. A |-> B ) )

Syntax hints:   -> wi 4   = wceq 1353   e. wcel 2148   {cab 2163   A.wral 2455   E.wrex 2456   |^|cint 3844   |^|_ciin 3887   |-> cmpt 4063   ran crn 4626

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4120  ax-pow 4173  ax-pr 4208

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-int 3845  df-iin 3889  df-br 4003  df-opab 4064  df-mpt 4065  df-cnv 4633  df-dm 4635  df-rn 4636

This theorem is referenced by:  dfiin3  4885  riinint  4886