Theorem f1fveq 4852

Description: Equality of function values for a one-to-one function.

Assertion

Ref	Expression
f1fveq	|- ((F:A-1-1->B /\ (C e. A /\ D e. A)) -> ((F` C) = (F` D) <-> C = D))

Proof of Theorem f1fveq

Step	Hyp	Ref	Expression
1		fveq2 4681	. . . . . . 7 |- (x = C -> (F` x) = (F` C))
2	1	eqeq1d 1892	. . . . . 6 |- (x = C -> ((F` x) = (F` y) <-> (F` C) = (F` y)))
3		eqeq1 1890	. . . . . 6 |- (x = C -> (x = y <-> C = y))
4	2, 3	imbi12d 688	. . . . 5 |- (x = C -> (((F` x) = (F` y) -> x = y) <-> ((F` C) = (F` y) -> C = y)))
5	4	imbi2d 674	. . . 4 |- (x = C -> ((F:A-1-1->B -> ((F` x) = (F` y) -> x = y)) <-> (F:A-1-1->B -> ((F` C) = (F` y) -> C = y))))
6		fveq2 4681	. . . . . . 7 |- (y = D -> (F` y) = (F` D))
7	6	eqeq2d 1895	. . . . . 6 |- (y = D -> ((F` C) = (F` y) <-> (F` C) = (F` D)))
8		eqeq2 1893	. . . . . 6 |- (y = D -> (C = y <-> C = D))
9	7, 8	imbi12d 688	. . . . 5 |- (y = D -> (((F` C) = (F` y) -> C = y) <-> ((F` C) = (F` D) -> C = D)))
10	9	imbi2d 674	. . . 4 |- (y = D -> ((F:A-1-1->B -> ((F` C) = (F` y) -> C = y)) <-> (F:A-1-1->B -> ((F` C) = (F` D) -> C = D))))
11		dff13 4850	. . . . . . 7 |- (F:A-1-1->B <-> (F:A-->B /\ A.x e. A A.y e. A ((F` x) = (F` y) -> x = y)))
12	11	simprbi 353	. . . . . 6 |- (F:A-1-1->B -> A.x e. A A.y e. A ((F` x) = (F` y) -> x = y))
13		ra42 2157	. . . . . 6 |- (A.x e. A A.y e. A ((F` x) = (F` y) -> x = y) -> ((x e. A /\ y e. A) -> ((F` x) = (F` y) -> x = y)))
14	12, 13	syl 12	. . . . 5 |- (F:A-1-1->B -> ((x e. A /\ y e. A) -> ((F` x) = (F` y) -> x = y)))
15	14	com12 14	. . . 4 |- ((x e. A /\ y e. A) -> (F:A-1-1->B -> ((F` x) = (F` y) -> x = y)))
16	5, 10, 15	vtocl2ga 2353	. . 3 |- ((C e. A /\ D e. A) -> (F:A-1-1->B -> ((F` C) = (F` D) -> C = D)))
17	16	impcom 378	. 2 |- ((F:A-1-1->B /\ (C e. A /\ D e. A)) -> ((F` C) = (F` D) -> C = D))
18		fveq2 4681	. 2 |- (C = D -> (F` C) = (F` D))
19	17, 18	impbid1 575	1 |- ((F:A-1-1->B /\ (C e. A /\ D e. A)) -> ((F` C) = (F` D) <-> C = D))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  A.wral 2105  -->wf 3994  -1-1->wf1 3995  ` cfv 3998

This theorem is referenced by:  isowe 4880  f1oiso 4881  f1oweALT 4883  2dom 5486  xpdom2 5501  ac6sfilem3 5508  mapenlem2 5584  hartoglem 5692  unidom 5970  eff1i 10098  njtlc 14389  gaplc 14731  grpdlcan 14739  grpdivzer 14740  hartoglemOLD 15383  f1elima 15719  metf1o 15843  rngisocnv 16135

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

Copyright terms: Public domain