Theorem ffnfvf 3835

Description: A function maps to a class to which all values belong. This version of ffnfv 3834 uses bound-variable hypotheses instead of distinct variable conditions.

Hypotheses

Ref	Expression
ffnfvf.1	|- (y e. A -> A.x y e. A)
ffnfvf.2	|- (y e. B -> A.x y e. B)
ffnfvf.3	|- (y e. F -> A.x y e. F)

Assertion

Ref	Expression
ffnfvf	|- (F:A-->B <-> (F Fn A /\ A.x e. A (F` x) e. B))

Distinct variable groups:   y,A   y,B   y,F   x,y

Proof of Theorem ffnfvf

Step	Hyp	Ref	Expression
1		ffnfv 3834	. 2 |- (F:A-->B <-> (F Fn A /\ A.z e. A (F` z) e. B))
2		ax-17 973	. . . 4 |- (y e. A -> A.z y e. A)
3		ffnfvf.1	. . . 4 |- (y e. A -> A.x y e. A)
4		ffnfvf.3	. . . . . 6 |- (y e. F -> A.x y e. F)
5		ax-17 973	. . . . . 6 |- (y e. z -> A.x y e. z)
6	4, 5	hbfv 3735	. . . . 5 |- (y e. (F` z) -> A.x y e. (F` z))
7		ffnfvf.2	. . . . 5 |- (y e. B -> A.x y e. B)
8	6, 7	hbel 1569	. . . 4 |- ((F` z) e. B -> A.x(F` z) e. B)
9		ax-17 973	. . . 4 |- ((F` x) e. B -> A.z(F` x) e. B)
10		fveq2 3730	. . . . 5 |- (z = x -> (F` z) = (F` x))
11	10	eleq1d 1543	. . . 4 |- (z = x -> ((F` z) e. B <-> (F` x) e. B))
12	2, 3, 8, 9, 11	cbvralf 1799	. . 3 |- (A.z e. A (F` z) e. B <-> A.x e. A (F` x) e. B)
13	12	anbi2i 482	. 2 |- ((F Fn A /\ A.z e. A (F` z) e. B) <-> (F Fn A /\ A.x e. A (F` x) e. B))
14	1, 13	bitr 173	1 |- (F:A-->B <-> (F Fn A /\ A.x e. A (F` x) e. B))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223  A.wal 956   = wceq 958   e. wcel 960  A.wral 1648   Fn wfn 3183  -->wf 3184  ` cfv 3188

This theorem is referenced by:  ixpf 4362

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-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-f 3200  df-fv 3204

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