Theorem f1fvf 2917

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 group(s):   x,y,A   x,B,y   z,F   x,z,y

Proof of Theorem f1fvf

Step	Hyp	Ref	Expression
1		f1fv 2916	. 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-17 925	. . . . . . . . 9 |- (z e. w -> A.y z e. w)
4	2, 3	hbfv 2837	. . . . . . . 8 |- (z e. (F` w) -> A.y z e. (F` w))
5		ax-17 925	. . . . . . . . 9 |- (z e. v -> A.y z e. v)
6	2, 5	hbfv 2837	. . . . . . . 8 |- (z e. (F` v) -> A.y z e. (F` v))
7	4, 6	hbeq 1171	. . . . . . 7 |- ((F` w) = (F` v) -> A.y(F` w) = (F` v))
8		ax-17 925	. . . . . . 7 |- (w = v -> A.y w = v)
9	7, 8	hbim 702	. . . . . 6 |- (((F` w) = (F` v) -> w = v) -> A.y((F` w) = (F` v) -> w = v))
10		ax-17 925	. . . . . . 7 |- ((F` w) = (F` y) -> A.v(F` w) = (F` y))
11		ax-17 925	. . . . . . 7 |- (w = y -> A.v w = y)
12	10, 11	hbim 702	. . . . . 6 |- (((F` w) = (F` y) -> w = y) -> A.v((F` w) = (F` y) -> w = y))
13		fveq2 2832	. . . . . . . 8 |- (v = y -> (F` v) = (F` y))
14	13	cleq2d 1112	. . . . . . 7 |- (v = y -> ((F` w) = (F` v) <-> (F` w) = (F` y)))
15		cleq2 1110	. . . . . . 7 |- (v = y -> (w = v <-> w = y))
16	14, 15	imbi12d 474	. . . . . 6 |- (v = y -> (((F` w) = (F` v) -> w = v) <-> ((F` w) = (F` y) -> w = y)))
17	9, 12, 16	cbvral 1331	. . . . 5 |- (A.v e. A ((F` w) = (F` v) -> w = v) <-> A.y e. A ((F` w) = (F` y) -> w = y))
18	17	biral 1223	. . . 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-17 925	. . . . . 6 |- (y e. A -> A.x y e. A)
20		f1fvf.1	. . . . . . . . 9 |- (z e. F -> A.x z e. F)
21		ax-17 925	. . . . . . . . 9 |- (z e. w -> A.x z e. w)
22	20, 21	hbfv 2837	. . . . . . . 8 |- (z e. (F` w) -> A.x z e. (F` w))
23		ax-17 925	. . . . . . . . 9 |- (z e. y -> A.x z e. y)
24	20, 23	hbfv 2837	. . . . . . . 8 |- (z e. (F` y) -> A.x z e. (F` y))
25	22, 24	hbeq 1171	. . . . . . 7 |- ((F` w) = (F` y) -> A.x(F` w) = (F` y))
26		ax-17 925	. . . . . . 7 |- (w = y -> A.x w = y)
27	25, 26	hbim 702	. . . . . 6 |- (((F` w) = (F` y) -> w = y) -> A.x((F` w) = (F` y) -> w = y))
28	19, 27	hbral 1236	. . . . 5 |- (A.y e. A ((F` w) = (F` y) -> w = y) -> A.xA.y e. A ((F` w) = (F` y) -> w = y))
29		ax-17 925	. . . . 5 |- (A.y e. A ((F` x) = (F` y) -> x = y) -> A.wA.y e. A ((F` x) = (F` y) -> x = y))
30		fveq2 2832	. . . . . . . 8 |- (w = x -> (F` w) = (F` x))
31	30	cleq1d 1109	. . . . . . 7 |- (w = x -> ((F` w) = (F` y) <-> (F` x) = (F` y)))
32		cleq1 1107	. . . . . . 7 |- (w = x -> (w = y <-> x = y))
33	31, 32	imbi12d 474	. . . . . 6 |- (w = x -> (((F` w) = (F` y) -> w = y) <-> ((F` x) = (F` y) -> x = y)))
34	33	biraldv 1219	. . . . 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 1331	. . . 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 151	. . 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 367	. 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 151	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 2   <-> wb 127   /\ wa 196  A.wal 672   = weq 797   e. wel 803   = wceq 1091   e. wcel 1092  A.wral 1201  -->wf 2418  -1-1->wf1 2419  ` cfv 2422

This theorem is referenced by:  dom2d 3307

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-id 2125  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fv 2438

metamath.org