Theorem dff13 4850

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

Assertion

Ref	Expression
dff13	|- (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   x,F,y

Proof of Theorem dff13

Step	Hyp	Ref	Expression
1		dff12 4609	. 2 |- (F:A-1-1->B <-> (F:A-->B /\ A.zE*x xFz))
2		ffn 4562	. . . 4 |- (F:A-->B -> F Fn A)
3		fndm 4512	. . . . . . . . . . . . . . 15 |- (F Fn A -> dom F = A)
4	3	eleq2d 1964	. . . . . . . . . . . . . 14 |- (F Fn A -> (x e. dom F <-> x e. A))
5		visset 2295	. . . . . . . . . . . . . . 15 |- x e. _V
6	5	breldm 4161	. . . . . . . . . . . . . 14 |- (xFz -> x e. dom F)
7	4, 6	syl5bi 225	. . . . . . . . . . . . 13 |- (F Fn A -> (xFz -> x e. A))
8	3	eleq2d 1964	. . . . . . . . . . . . . 14 |- (F Fn A -> (y e. dom F <-> y e. A))
9		visset 2295	. . . . . . . . . . . . . . 15 |- y e. _V
10	9	breldm 4161	. . . . . . . . . . . . . 14 |- (yFz -> y e. dom F)
11	8, 10	syl5bi 225	. . . . . . . . . . . . 13 |- (F Fn A -> (yFz -> y e. A))
12	7, 11	anim12d 617	. . . . . . . . . . . 12 |- (F Fn A -> ((xFz /\ yFz) -> (x e. A /\ y e. A)))
13	12	pm4.71rd 701	. . . . . . . . . . 11 |- (F Fn A -> ((xFz /\ yFz) <-> ((x e. A /\ y e. A) /\ (xFz /\ yFz))))
14		visset 2295	. . . . . . . . . . . . . . . 16 |- z e. _V
15	14	fnbrfvb 4712	. . . . . . . . . . . . . . 15 |- ((F Fn A /\ x e. A) -> ((F` x) = z <-> xFz))
16		eqcom 1886	. . . . . . . . . . . . . . 15 |- (z = (F` x) <-> (F` x) = z)
17	15, 16	syl5bb 591	. . . . . . . . . . . . . 14 |- ((F Fn A /\ x e. A) -> (z = (F` x) <-> xFz))
18	14	fnbrfvb 4712	. . . . . . . . . . . . . . 15 |- ((F Fn A /\ y e. A) -> ((F` y) = z <-> yFz))
19		eqcom 1886	. . . . . . . . . . . . . . 15 |- (z = (F` y) <-> (F` y) = z)
20	18, 19	syl5bb 591	. . . . . . . . . . . . . 14 |- ((F Fn A /\ y e. A) -> (z = (F` y) <-> yFz))
21	17, 20	bi2anan9 694	. . . . . . . . . . . . 13 |- (((F Fn A /\ x e. A) /\ (F Fn A /\ y e. A)) -> ((z = (F` x) /\ z = (F` y)) <-> (xFz /\ yFz)))
22	21	anandis 570	. . . . . . . . . . . 12 |- ((F Fn A /\ (x e. A /\ y e. A)) -> ((z = (F` x) /\ z = (F` y)) <-> (xFz /\ yFz)))
23	22	pm5.32da 711	. . . . . . . . . . 11 |- (F Fn A -> (((x e. A /\ y e. A) /\ (z = (F` x) /\ z = (F` y))) <-> ((x e. A /\ y e. A) /\ (xFz /\ yFz))))
24	13, 23	bitr4d 590	. . . . . . . . . 10 |- (F Fn A -> ((xFz /\ yFz) <-> ((x e. A /\ y e. A) /\ (z = (F` x) /\ z = (F` y)))))
25	24	imbi1d 675	. . . . . . . . 9 |- (F Fn A -> (((xFz /\ yFz) -> x = y) <-> (((x e. A /\ y e. A) /\ (z = (F` x) /\ z = (F` y))) -> x = y)))
26		impexp 374	. . . . . . . . 9 |- ((((x e. A /\ y e. A) /\ (z = (F` x) /\ z = (F` y))) -> x = y) <-> ((x e. A /\ y e. A) -> ((z = (F` x) /\ z = (F` y)) -> x = y)))
27	25, 26	syl6bb 595	. . . . . . . 8 |- (F Fn A -> (((xFz /\ yFz) -> x = y) <-> ((x e. A /\ y e. A) -> ((z = (F` x) /\ z = (F` y)) -> x = y))))
28	27	albidv 1656	. . . . . . 7 |- (F Fn A -> (A.z((xFz /\ yFz) -> x = y) <-> A.z((x e. A /\ y e. A) -> ((z = (F` x) /\ z = (F` y)) -> x = y))))
29		19.21v 1663	. . . . . . . 8 |- (A.z((x e. A /\ y e. A) -> ((z = (F` x) /\ z = (F` y)) -> x = y)) <-> ((x e. A /\ y e. A) -> A.z((z = (F` x) /\ z = (F` y)) -> x = y)))
30		19.23v 1672	. . . . . . . . . 10 |- (A.z((z = (F` x) /\ z = (F` y)) -> x = y) <-> (E.z(z = (F` x) /\ z = (F` y)) -> x = y))
31		fvex 4689	. . . . . . . . . . . 12 |- (F` x) e. _V
32	31	eqvinc 2387	. . . . . . . . . . 11 |- ((F` x) = (F` y) <-> E.z(z = (F` x) /\ z = (F` y)))
33	32	imbi1i 203	. . . . . . . . . 10 |- (((F` x) = (F` y) -> x = y) <-> (E.z(z = (F` x) /\ z = (F` y)) -> x = y))
34	30, 33	bitr4i 193	. . . . . . . . 9 |- (A.z((z = (F` x) /\ z = (F` y)) -> x = y) <-> ((F` x) = (F` y) -> x = y))
35	34	imbi2i 202	. . . . . . . 8 |- (((x e. A /\ y e. A) -> A.z((z = (F` x) /\ z = (F` y)) -> x = y)) <-> ((x e. A /\ y e. A) -> ((F` x) = (F` y) -> x = y)))
36	29, 35	bitri 190	. . . . . . 7 |- (A.z((x e. A /\ y e. A) -> ((z = (F` x) /\ z = (F` y)) -> x = y)) <-> ((x e. A /\ y e. A) -> ((F` x) = (F` y) -> x = y)))
37	28, 36	syl6bb 595	. . . . . 6 |- (F Fn A -> (A.z((xFz /\ yFz) -> x = y) <-> ((x e. A /\ y e. A) -> ((F` x) = (F` y) -> x = y))))
38	37	2albidv 1658	. . . . 5 |- (F Fn A -> (A.xA.yA.z((xFz /\ yFz) -> x = y) <-> A.xA.y((x e. A /\ y e. A) -> ((F` x) = (F` y) -> x = y))))
39		breq1 3341	. . . . . . . 8 |- (x = y -> (xFz <-> yFz))
40	39	mo4 1799	. . . . . . 7 |- (E*x xFz <-> A.xA.y((xFz /\ yFz) -> x = y))
41	40	albii 1346	. . . . . 6 |- (A.zE*x xFz <-> A.zA.xA.y((xFz /\ yFz) -> x = y))
42		alcom 1379	. . . . . 6 |- (A.zA.xA.y((xFz /\ yFz) -> x = y) <-> A.xA.zA.y((xFz /\ yFz) -> x = y))
43		alcom 1379	. . . . . . 7 |- (A.zA.y((xFz /\ yFz) -> x = y) <-> A.yA.z((xFz /\ yFz) -> x = y))
44	43	albii 1346	. . . . . 6 |- (A.xA.zA.y((xFz /\ yFz) -> x = y) <-> A.xA.yA.z((xFz /\ yFz) -> x = y))
45	41, 42, 44	3bitri 194	. . . . 5 |- (A.zE*x xFz <-> A.xA.yA.z((xFz /\ yFz) -> x = y))
46		r2al 2136	. . . . 5 |- (A.x e. A A.y e. A ((F` x) = (F` y) -> x = y) <-> A.xA.y((x e. A /\ y e. A) -> ((F` x) = (F` y) -> x = y)))
47	38, 45, 46	3bitr4g 614	. . . 4 |- (F Fn A -> (A.zE*x xFz <-> A.x e. A A.y e. A ((F` x) = (F` y) -> x = y)))
48	2, 47	syl 12	. . 3 |- (F:A-->B -> (A.zE*x xFz <-> A.x e. A A.y e. A ((F` x) = (F` y) -> x = y)))
49	48	pm5.32i 707	. 2 |- ((F:A-->B /\ A.zE*x xFz) <-> (F:A-->B /\ A.x e. A A.y e. A ((F` x) = (F` y) -> x = y)))
50	1, 49	bitri 190	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 163   /\ wa 240  A.wal 1296   = wceq 1298   e. wcel 1300  E.wex 1326  E*wmo 1772  A.wral 2105   class class class wbr 3338  dom cdm 3986   Fn wfn 3993  -->wf 3994  -1-1->wf1 3995  ` cfv 3998

This theorem is referenced by:  dff13f 4851  f1fveq 4852  dff1o6 4853  tz7.48lem 5164  omsmo 5314  mapenlem2 5584  unfilem2 5642  inf3lem6 5724  alephiso 6040  icoshftf1oii 7578  om2uzf1oi 7712  reeff1 8675  grplactf1o 9406  efif1 10091  eff1i 10098  pjmf1 11296  unopf1o 11477  ghomf1olem 13637  injrec 14461  injsurinj 14487  homcard 14893  trnij 15015  f1opr 15714  grpkerinj 16042

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fv 4014

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