Theorem ffnfvf 3829

Description: A function maps to a class to which all values belong. This version of ffnfv 3828 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 3828	. 2 |- (F:A-->B <-> (F Fn A /\ A.z e. A (F` z) e. B))
2		ax-17 971	. . . 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-17 971	. . . . . 6 |- (y e. z -> A.x y e. z)
6	4, 5	hbfv 3729	. . . . 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 1566	. . . 4 |- ((F` z) e. B -> A.x(F` z) e. B)
9		ax-17 971	. . . 4 |- ((F` x) e. B -> A.z(F` x) e. B)
10		fveq2 3724	. . . . 5 |- (z = x -> (F` z) = (F` x))
11	10	eleq1d 1540	. . . 4 |- (z = x -> ((F` z) e. B <-> (F` x) e. B))
12	2, 3, 8, 9, 11	cbvralf 1796	. . 3 |- (A.z e. A (F` z) e. B <-> A.x e. A (F` x) e. B)
13	12	anbi2i 480	. 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 954   = wceq 956   e. wcel 958  A.wral 1645   Fn wfn 3177  -->wf 3178  ` cfv 3182

This theorem is referenced by:  ixpf 4356

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-fv 3198

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