Theorem eqfnfvf 3793

Description: Equality of functions is determined by their values. Exercise 4 of [TakeutiZaring] p. 28. This version of eqfnfv 3792 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 3792	. 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-17 970	. . . . . 6 |- (y e. z -> A.x y e. z)
4	2, 3	hbfv 3724	. . . . 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 3724	. . . . 5 |- (y e. (G` z) -> A.x y e. (G` z))
7	4, 6	hbeq 1563	. . . 4 |- ((F` z) = (G` z) -> A.x(F` z) = (G` z))
8		ax-17 970	. . . 4 |- ((F` x) = (G` x) -> A.z(F` x) = (G` x))
9		fveq2 3719	. . . . 5 |- (z = x -> (F` z) = (F` x))
10		fveq2 3719	. . . . 5 |- (z = x -> (G` z) = (G` x))
11	9, 10	eqeq12d 1487	. . . 4 |- (z = x -> ((F` z) = (G` z) <-> (F` x) = (G` x)))
12	7, 8, 11	cbvral 1795	. . 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 535	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 953   = wceq 955   e. wcel 957  A.wral 1643   Fn wfn 3173  ` cfv 3178

This theorem is referenced by:  fopabco 3827  fopabcos 3828  fopabsn 3835  pw2en 4435  cnvtr 10554

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458  ax-sep 2699  ax-pow 2738  ax-pr 2775

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-ral 1647  df-rex 1648  df-v 1809  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-nul 2278  df-pw 2399  df-sn 2409  df-pr 2410  df-op 2413  df-uni 2500  df-br 2616  df-opab 2663  df-id 2831  df-xp 3180  df-rel 3181  df-cnv 3182  df-co 3183  df-dm 3184  df-rn 3185  df-res 3186  df-ima 3187  df-fun 3188  df-fn 3189  df-fv 3194

Copyright terms: Public domain