Theorem fniinfv 5713

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 5665	. . . . 5 |- ( ( Fun F /\ x e. dom F ) -> ( F ` x ) e. _V )
2	1	funfni 5439	. . . 4 |- ( ( F Fn A /\ x e. A ) -> ( F ` x ) e. _V )
3	2	ralrimiva 2606	. . 3 |- ( F Fn A -> A. x e. A ( F ` x ) e. _V )
4		dfiin2g 4008	. . 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 5701	. . 3 |- ( F Fn A -> ran F = { y | E. x e. A y = ( F ` x ) } )
7	6	inteqd 3938	. 2 |- ( F Fn A -> |^| ran F = |^| { y | E. x e. A y = ( F ` x ) } )
8	5, 7	eqtr4d 2267	1 |- ( F Fn A -> |^|_ x e. A ( F ` x ) = |^| ran F )

Syntax hints:   -> wi 4   = wceq 1398   e. wcel 2202   {cab 2217   A.wral 2511   E.wrex 2512   _Vcvv 2803   |^|cint 3933   |^|_ciin 3976   ran crn 4732   Fn wfn 5328   ` cfv 5333

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 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-sbc 3033  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iin 3978  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-iota 5293  df-fun 5335  df-fn 5336  df-fv 5341

This theorem is referenced by: (None)