Theorem fniinfv 5525

Description: The indexed intersection of a function's values is the intersection of its range. (Contributed by NM, 20-Oct-2005.)

Assertion

Ref	Expression
fniinfv	|- ( F Fn A -> |^|_ x e. A ( F ` x ) = |^| ran F )

Distinct variable groups:   x, A   x, F

Proof of Theorem fniinfv

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		funfvex 5484	. . . . 5 |- ( ( Fun F /\ x e. dom F ) -> ( F ` x ) e. _V )
2	1	funfni 5269	. . . 4 |- ( ( F Fn A /\ x e. A ) -> ( F ` x ) e. _V )
3	2	ralrimiva 2530	. . 3 |- ( F Fn A -> A. x e. A ( F ` x ) e. _V )
4		dfiin2g 3882	. . 3 |- ( A. x e. A ( F ` x ) e. _V -> |^|_ x e. A ( F ` x ) = |^| { y | E. x e. A y = ( F ` x ) } )
5	3, 4	syl 14	. 2 |- ( F Fn A -> |^|_ x e. A ( F ` x ) = |^| { y | E. x e. A y = ( F ` x ) } )
6		fnrnfv 5514	. . 3 |- ( F Fn A -> ran F = { y | E. x e. A y = ( F ` x ) } )
7	6	inteqd 3812	. 2 |- ( F Fn A -> |^| ran F = |^| { y | E. x e. A y = ( F ` x ) } )
8	5, 7	eqtr4d 2193	1 |- ( F Fn A -> |^|_ x e. A ( F ` x ) = |^| ran F )

Syntax hints:   -> wi 4   = wceq 1335   e. wcel 2128   {cab 2143   A.wral 2435   E.wrex 2436   _Vcvv 2712   |^|cint 3807   |^|_ciin 3850   ran crn 4586   Fn wfn 5164   ` cfv 5169

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-sbc 2938  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-int 3808  df-iin 3852  df-br 3966  df-opab 4026  df-mpt 4027  df-id 4253  df-xp 4591  df-rel 4592  df-cnv 4593  df-co 4594  df-dm 4595  df-rn 4596  df-iota 5134  df-fun 5171  df-fn 5172  df-fv 5177

This theorem is referenced by: (None)