Theorem f1fveq 2918

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 2832	. . . . . . . 8 |- (x = C -> (F` x) = (F` C))
2	1	cleq1d 1109	. . . . . . 7 |- (x = C -> ((F` x) = (F` y) <-> (F` C) = (F` y)))
3		cleq1 1107	. . . . . . 7 |- (x = C -> (x = y <-> C = y))
4	2, 3	imbi12d 474	. . . . . 6 |- (x = C -> (((F` x) = (F` y) -> x = y) <-> ((F` C) = (F` y) -> C = y)))
5	4	imbi2d 464	. . . . 5 |- (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 2832	. . . . . . . 8 |- (y = D -> (F` y) = (F` D))
7	6	cleq2d 1112	. . . . . . 7 |- (y = D -> ((F` C) = (F` y) <-> (F` C) = (F` D)))
8		cleq2 1110	. . . . . . 7 |- (y = D -> (C = y <-> C = D))
9	7, 8	imbi12d 474	. . . . . 6 |- (y = D -> (((F` C) = (F` y) -> C = y) <-> ((F` C) = (F` D) -> C = D)))
10	9	imbi2d 464	. . . . 5 |- (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 2916	. . . . . . . 8 |- (F:A-1-1->B <-> (F:A-->B /\ A.x e. A A.y e. A ((F` x) = (F` y) -> x = y)))
12	11	pm3.27bd 263	. . . . . . 7 |- (F:A-1-1->B -> A.x e. A A.y e. A ((F` x) = (F` y) -> x = y))
13		ra42 1245	. . . . . . 7 |- (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	. . . . . 6 |- (F:A-1-1->B -> ((x e. A /\ y e. A) -> ((F` x) = (F` y) -> x = y)))
15	14	com12 13	. . . . 5 |- ((x e. A /\ y e. A) -> (F:A-1-1->B -> ((F` x) = (F` y) -> x = y)))
16	5, 10, 15	vtocl2ga 1388	. . . 4 |- ((C e. A /\ D e. A) -> (F:A-1-1->B -> ((F` C) = (F` D) -> C = D)))
17	16	com12 13	. . 3 |- (F:A-1-1->B -> ((C e. A /\ D e. A) -> ((F` C) = (F` D) -> C = D)))
18	17	imp 277	. 2 |- ((F:A-1-1->B /\ (C e. A /\ D e. A)) -> ((F` C) = (F` D) -> C = D))
19		fveq2 2832	. . 3 |- (C = D -> (F` C) = (F` D))
20	19	a1i 7	. 2 |- ((F:A-1-1->B /\ (C e. A /\ D e. A)) -> (C = D -> (F` C) = (F` D)))
21	18, 20	impbid 397	1 |- ((F:A-1-1->B /\ (C e. A /\ D e. A)) -> ((F` C) = (F` D) <-> C = D))

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196   = weq 797   = wceq 1091   e. wcel 1092  A.wral 1201  -->wf 2418  -1-1->wf1 2419  ` cfv 2422

This theorem is referenced by:  isowe 2941  f1oiso 2942  f1oweOLD 2944  2dom 3332  xpdom2 3345  mapenlem2 3385

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

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