Theorem fniinfv 5639

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 5595	. . . . 5 |- ( ( Fun F /\ x e. dom F ) -> ( F ` x ) e. _V )
2	1	funfni 5377	. . . 4 |- ( ( F Fn A /\ x e. A ) -> ( F ` x ) e. _V )
3	2	ralrimiva 2579	. . 3 |- ( F Fn A -> A. x e. A ( F ` x ) e. _V )
4		dfiin2g 3960	. . 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 5627	. . 3 |- ( F Fn A -> ran F = { y | E. x e. A y = ( F ` x ) } )
7	6	inteqd 3890	. 2 |- ( F Fn A -> |^| ran F = |^| { y | E. x e. A y = ( F ` x ) } )
8	5, 7	eqtr4d 2241	1 |- ( F Fn A -> |^|_ x e. A ( F ` x ) = |^| ran F )

Syntax hints:   -> wi 4   = wceq 1373   e. wcel 2176   {cab 2191   A.wral 2484   E.wrex 2485   _Vcvv 2772   |^|cint 3885   |^|_ciin 3928   ran crn 4677   Fn wfn 5267   ` cfv 5272

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4163  ax-pow 4219  ax-pr 4254

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-sbc 2999  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iin 3930  df-br 4046  df-opab 4107  df-mpt 4108  df-id 4341  df-xp 4682  df-rel 4683  df-cnv 4684  df-co 4685  df-dm 4686  df-rn 4687  df-iota 5233  df-fun 5274  df-fn 5275  df-fv 5280

This theorem is referenced by: (None)