Theorem f1fveq 3882

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 3730	. . . . . . 7 |- (x = C -> (F` x) = (F` C))
2	1	eqeq1d 1486	. . . . . 6 |- (x = C -> ((F` x) = (F` y) <-> (F` C) = (F` y)))
3		eqeq1 1484	. . . . . 6 |- (x = C -> (x = y <-> C = y))
4	2, 3	imbi12d 628	. . . . 5 |- (x = C -> (((F` x) = (F` y) -> x = y) <-> ((F` C) = (F` y) -> C = y)))
5	4	imbi2d 614	. . . 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 3730	. . . . . . 7 |- (y = D -> (F` y) = (F` D))
7	6	eqeq2d 1489	. . . . . 6 |- (y = D -> ((F` C) = (F` y) <-> (F` C) = (F` D)))
8		eqeq2 1487	. . . . . 6 |- (y = D -> (C = y <-> C = D))
9	7, 8	imbi12d 628	. . . . 5 |- (y = D -> (((F` C) = (F` y) -> C = y) <-> ((F` C) = (F` D) -> C = D)))
10	9	imbi2d 614	. . . 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		f1fv 3880	. . . . . . 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	pm3.27bi 326	. . . . . 6 |- (F:A-1-1->B -> A.x e. A A.y e. A ((F` x) = (F` y) -> x = y))
13		ra42 1699	. . . . . 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 10	. . . . 5 |- (F:A-1-1->B -> ((x e. A /\ y e. A) -> ((F` x) = (F` y) -> x = y)))
15	14	com12 11	. . . 4 |- ((x e. A /\ y e. A) -> (F:A-1-1->B -> ((F` x) = (F` y) -> x = y)))
16	5, 10, 15	vtocl2ga 1856	. . 3 |- ((C e. A /\ D e. A) -> (F:A-1-1->B -> ((F` C) = (F` D) -> C = D)))
17	16	impcom 351	. 2 |- ((F:A-1-1->B /\ (C e. A /\ D e. A)) -> ((F` C) = (F` D) -> C = D))
18		fveq2 3730	. 2 |- (C = D -> (F` C) = (F` D))
19	17, 18	impbid1 519	1 |- ((F:A-1-1->B /\ (C e. A /\ D e. A)) -> ((F` C) = (F` D) <-> C = D))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 958   e. wcel 960  A.wral 1648  -->wf 3184  -1-1->wf1 3185  ` cfv 3188

This theorem is referenced by:  isowe 3909  f1oiso 3910  f1oweALT 3912  2dom 4433  xpdom2 4448  mapenlem2 4496  unidom 4818  eff1i 8739

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-f1 3201  df-fv 3204

Copyright terms: Public domain