Theorem fniinfv 5660

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 5616	. . . . 5 |- ( ( Fun F /\ x e. dom F ) -> ( F ` x ) e. _V )
2	1	funfni 5395	. . . 4 |- ( ( F Fn A /\ x e. A ) -> ( F ` x ) e. _V )
3	2	ralrimiva 2581	. . 3 |- ( F Fn A -> A. x e. A ( F ` x ) e. _V )
4		dfiin2g 3974	. . 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 5648	. . 3 |- ( F Fn A -> ran F = { y | E. x e. A y = ( F ` x ) } )
7	6	inteqd 3904	. 2 |- ( F Fn A -> |^| ran F = |^| { y | E. x e. A y = ( F ` x ) } )
8	5, 7	eqtr4d 2243	1 |- ( F Fn A -> |^|_ x e. A ( F ` x ) = |^| ran F )

Syntax hints:   -> wi 4   = wceq 1373   e. wcel 2178   {cab 2193   A.wral 2486   E.wrex 2487   _Vcvv 2776   |^|cint 3899   |^|_ciin 3942   ran crn 4694   Fn wfn 5285   ` cfv 5290

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-sbc 3006  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-iin 3944  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-iota 5251  df-fun 5292  df-fn 5293  df-fv 5298

This theorem is referenced by: (None)