Theorem fniinfv 3772

Description: The indexed intersection of a function's values is the intersection of its range.

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

Step	Hyp	Ref	Expression
1		fnrnfv 3765	. . 3 |- (F Fn A -> ran F = {y | E.x e. A y = (F` x)})
2	1	inteqd 2542	. 2 |- (F Fn A -> |^|ran F = |^|{y | E.x e. A y = (F` x)})
3		fvex 3738	. . 3 |- (F` x) e. V
4	3	dfiin2 2592	. 2 |- |^|_x e. A (F` x) = |^|{y | E.x e. A y = (F` x)}
5	2, 4	syl6reqr 1529	1 |- (F Fn A -> |^|_x e. A (F` x) = |^|ran F)

Syntax hints:   -> wi 3   = wceq 958  {cab 1466  E.wrex 1649  |^|cint 2537  |^|_ciin 2571  ran crn 3177   Fn wfn 3183  ` cfv 3188

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-int 2538  df-iin 2573  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-fv 3204

Copyright terms: Public domain