Theorem f1fvf 3875

Description: A one-to-one function in terms of function values. Compare Theorem 4.8(iv) of [Monk1] p. 43.

Hypotheses

Ref	Expression
f1fvf.1	|- (z e. F -> A.x z e. F)
f1fvf.2	|- (z e. F -> A.y z e. F)

Assertion

Ref	Expression
f1fvf	|- (F:A-1-1->B <-> (F:A-->B /\ A.x e. A A.y e. A ((F` x) = (F` y) -> x = y)))

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

Proof of Theorem f1fvf

Step	Hyp	Ref	Expression
1		f1fv 3874	. 2 |- (F:A-1-1->B <-> (F:A-->B /\ A.w e. A A.v e. A ((F` w) = (F` v) -> w = v)))
2		f1fvf.2	. . . . . . . . 9 |- (z e. F -> A.y z e. F)
3		ax-17 971	. . . . . . . . 9 |- (z e. w -> A.y z e. w)
4	2, 3	hbfv 3729	. . . . . . . 8 |- (z e. (F` w) -> A.y z e. (F` w))
5		ax-17 971	. . . . . . . . 9 |- (z e. v -> A.y z e. v)
6	2, 5	hbfv 3729	. . . . . . . 8 |- (z e. (F` v) -> A.y z e. (F` v))
7	4, 6	hbeq 1565	. . . . . . 7 |- ((F` w) = (F` v) -> A.y(F` w) = (F` v))
8		ax-17 971	. . . . . . 7 |- (w = v -> A.y w = v)
9	7, 8	hbim 1007	. . . . . 6 |- (((F` w) = (F` v) -> w = v) -> A.y((F` w) = (F` v) -> w = v))
10		ax-17 971	. . . . . . 7 |- ((F` w) = (F` y) -> A.v(F` w) = (F` y))
11		ax-17 971	. . . . . . 7 |- (w = y -> A.v w = y)
12	10, 11	hbim 1007	. . . . . 6 |- (((F` w) = (F` y) -> w = y) -> A.v((F` w) = (F` y) -> w = y))
13		fveq2 3724	. . . . . . . 8 |- (v = y -> (F` v) = (F` y))
14	13	eqeq2d 1486	. . . . . . 7 |- (v = y -> ((F` w) = (F` v) <-> (F` w) = (F` y)))
15		eqeq2 1484	. . . . . . 7 |- (v = y -> (w = v <-> w = y))
16	14, 15	imbi12d 626	. . . . . 6 |- (v = y -> (((F` w) = (F` v) -> w = v) <-> ((F` w) = (F` y) -> w = y)))
17	9, 12, 16	cbvral 1798	. . . . 5 |- (A.v e. A ((F` w) = (F` v) -> w = v) <-> A.y e. A ((F` w) = (F` y) -> w = y))
18	17	ralbii 1667	. . . 4 |- (A.w e. A A.v e. A ((F` w) = (F` v) -> w = v) <-> A.w e. A A.y e. A ((F` w) = (F` y) -> w = y))
19		ax-17 971	. . . . . 6 |- (y e. A -> A.x y e. A)
20		f1fvf.1	. . . . . . . . 9 |- (z e. F -> A.x z e. F)
21		ax-17 971	. . . . . . . . 9 |- (z e. w -> A.x z e. w)
22	20, 21	hbfv 3729	. . . . . . . 8 |- (z e. (F` w) -> A.x z e. (F` w))
23		ax-17 971	. . . . . . . . 9 |- (z e. y -> A.x z e. y)
24	20, 23	hbfv 3729	. . . . . . . 8 |- (z e. (F` y) -> A.x z e. (F` y))
25	22, 24	hbeq 1565	. . . . . . 7 |- ((F` w) = (F` y) -> A.x(F` w) = (F` y))
26		ax-17 971	. . . . . . 7 |- (w = y -> A.x w = y)
27	25, 26	hbim 1007	. . . . . 6 |- (((F` w) = (F` y) -> w = y) -> A.x((F` w) = (F` y) -> w = y))
28	19, 27	hbral 1686	. . . . 5 |- (A.y e. A ((F` w) = (F` y) -> w = y) -> A.xA.y e. A ((F` w) = (F` y) -> w = y))
29		ax-17 971	. . . . 5 |- (A.y e. A ((F` x) = (F` y) -> x = y) -> A.wA.y e. A ((F` x) = (F` y) -> x = y))
30		fveq2 3724	. . . . . . . 8 |- (w = x -> (F` w) = (F` x))
31	30	eqeq1d 1483	. . . . . . 7 |- (w = x -> ((F` w) = (F` y) <-> (F` x) = (F` y)))
32		eqeq1 1481	. . . . . . 7 |- (w = x -> (w = y <-> x = y))
33	31, 32	imbi12d 626	. . . . . 6 |- (w = x -> (((F` w) = (F` y) -> w = y) <-> ((F` x) = (F` y) -> x = y)))
34	33	ralbidv 1663	. . . . 5 |- (w = x -> (A.y e. A ((F` w) = (F` y) -> w = y) <-> A.y e. A ((F` x) = (F` y) -> x = y)))
35	28, 29, 34	cbvral 1798	. . . 4 |- (A.w e. A A.y e. A ((F` w) = (F` y) -> w = y) <-> A.x e. A A.y e. A ((F` x) = (F` y) -> x = y))
36	18, 35	bitr 173	. . 3 |- (A.w e. A A.v e. A ((F` w) = (F` v) -> w = v) <-> A.x e. A A.y e. A ((F` x) = (F` y) -> x = y))
37	36	anbi2i 480	. 2 |- ((F:A-->B /\ A.w e. A A.v e. A ((F` w) = (F` v) -> w = v)) <-> (F:A-->B /\ A.x e. A A.y e. A ((F` x) = (F` y) -> x = y)))
38	1, 37	bitr 173	1 |- (F:A-1-1->B <-> (F:A-->B /\ A.x e. A A.y e. A ((F` x) = (F` y) -> x = y)))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223  A.wal 954   = wceq 956   e. wcel 958  A.wral 1645  -->wf 3178  -1-1->wf1 3179  ` cfv 3182

This theorem is referenced by:  dom2d 4404

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-9 965  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-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-f 3194  df-f1 3195  df-fv 3198

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