Theorem eqfnfvf 3798

Description: Equality of functions is determined by their values. Exercise 4 of [TakeutiZaring] p. 28. This version of eqfnfv 3797 uses bound-variable hypotheses instead of distinct variable conditions.

Hypotheses

Ref	Expression
eqfnfvf.1	|- (y e. F -> A.x y e. F)
eqfnfvf.2	|- (y e. G -> A.x y e. G)

Assertion

Ref	Expression
eqfnfvf	|- ((F Fn A /\ G Fn B) -> (F = G <-> (A = B /\ A.x e. A (F` x) = (G` x))))

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

Proof of Theorem eqfnfvf

Step	Hyp	Ref	Expression
1		eqfnfv 3797	. 2 |- ((F Fn A /\ G Fn B) -> (F = G <-> (A = B /\ A.z e. A (F` z) = (G` z))))
2		eqfnfvf.1	. . . . . 6 |- (y e. F -> A.x y e. F)
3		ax-17 971	. . . . . 6 |- (y e. z -> A.x y e. z)
4	2, 3	hbfv 3729	. . . . 5 |- (y e. (F` z) -> A.x y e. (F` z))
5		eqfnfvf.2	. . . . . 6 |- (y e. G -> A.x y e. G)
6	5, 3	hbfv 3729	. . . . 5 |- (y e. (G` z) -> A.x y e. (G` z))
7	4, 6	hbeq 1565	. . . 4 |- ((F` z) = (G` z) -> A.x(F` z) = (G` z))
8		ax-17 971	. . . 4 |- ((F` x) = (G` x) -> A.z(F` x) = (G` x))
9		fveq2 3724	. . . . 5 |- (z = x -> (F` z) = (F` x))
10		fveq2 3724	. . . . 5 |- (z = x -> (G` z) = (G` x))
11	9, 10	eqeq12d 1489	. . . 4 |- (z = x -> ((F` z) = (G` z) <-> (F` x) = (G` x)))
12	7, 8, 11	cbvral 1798	. . 3 |- (A.z e. A (F` z) = (G` z) <-> A.x e. A (F` x) = (G` x))
13	12	anbi2i 480	. 2 |- ((A = B /\ A.z e. A (F` z) = (G` z)) <-> (A = B /\ A.x e. A (F` x) = (G` x)))
14	1, 13	syl6bb 536	1 |- ((F Fn A /\ G Fn B) -> (F = G <-> (A = B /\ A.x e. A (F` x) = (G` x))))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223  A.wal 954   = wceq 956   e. wcel 958  A.wral 1645   Fn wfn 3177  ` cfv 3182

This theorem is referenced by:  fopabco 3832  fopabcos 3833  fopabsn 3840  pw2en 4446  cnvtr 10638

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

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-rel 3185  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-fv 3198

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