Theorem njtlc 14389

Description: An injection is left-cancelable.

Assertion

Ref	Expression
njtlc	|- ((F:B-1-1->C /\ H:A-->B /\ K:A-->B) -> ((F o. H) = (F o. K) -> H = K))

Proof of Theorem njtlc

Step	Hyp	Ref	Expression
1		f1f 4610	. . . . . . . . . 10 |- (F:B-1-1->C -> F:B-->C)
2		ffun 4565	. . . . . . . . . 10 |- (F:B-->C -> Fun F)
3	1, 2	syl 12	. . . . . . . . 9 |- (F:B-1-1->C -> Fun F)
4	3	3ad2ant1 897	. . . . . . . 8 |- ((F:B-1-1->C /\ H:A-->B /\ K:A-->B) -> Fun F)
5	4	adantr 425	. . . . . . 7 |- (((F:B-1-1->C /\ H:A-->B /\ K:A-->B) /\ x e. A) -> Fun F)
6		simpl2 880	. . . . . . 7 |- (((F:B-1-1->C /\ H:A-->B /\ K:A-->B) /\ x e. A) -> H:A-->B)
7		simpr 350	. . . . . . 7 |- (((F:B-1-1->C /\ H:A-->B /\ K:A-->B) /\ x e. A) -> x e. A)
8		fvco3 4739	. . . . . . 7 |- ((Fun F /\ H:A-->B /\ x e. A) -> ((F o. H)` x) = (F` (H` x)))
9	5, 6, 7, 8	syl111anc 1100	. . . . . 6 |- (((F:B-1-1->C /\ H:A-->B /\ K:A-->B) /\ x e. A) -> ((F o. H)` x) = (F` (H` x)))
10		simpl3 881	. . . . . . 7 |- (((F:B-1-1->C /\ H:A-->B /\ K:A-->B) /\ x e. A) -> K:A-->B)
11		fvco3 4739	. . . . . . 7 |- ((Fun F /\ K:A-->B /\ x e. A) -> ((F o. K)` x) = (F` (K` x)))
12	5, 10, 7, 11	syl111anc 1100	. . . . . 6 |- (((F:B-1-1->C /\ H:A-->B /\ K:A-->B) /\ x e. A) -> ((F o. K)` x) = (F` (K` x)))
13	9, 12	eqeq12d 1899	. . . . 5 |- (((F:B-1-1->C /\ H:A-->B /\ K:A-->B) /\ x e. A) -> (((F o. H)` x) = ((F o. K)` x) <-> (F` (H` x)) = (F` (K` x))))
14		simpl1 879	. . . . . 6 |- (((F:B-1-1->C /\ H:A-->B /\ K:A-->B) /\ x e. A) -> F:B-1-1->C)
15		ffvelrn 4787	. . . . . . 7 |- ((H:A-->B /\ x e. A) -> (H` x) e. B)
16	15	3ad2antl2 1039	. . . . . 6 |- (((F:B-1-1->C /\ H:A-->B /\ K:A-->B) /\ x e. A) -> (H` x) e. B)
17		ffvelrn 4787	. . . . . . 7 |- ((K:A-->B /\ x e. A) -> (K` x) e. B)
18	17	3ad2antl3 1040	. . . . . 6 |- (((F:B-1-1->C /\ H:A-->B /\ K:A-->B) /\ x e. A) -> (K` x) e. B)
19		f1fveq 4852	. . . . . . 7 |- ((F:B-1-1->C /\ ((H` x) e. B /\ (K` x) e. B)) -> ((F` (H` x)) = (F` (K` x)) <-> (H` x) = (K` x)))
20	19	biimpd 170	. . . . . 6 |- ((F:B-1-1->C /\ ((H` x) e. B /\ (K` x) e. B)) -> ((F` (H` x)) = (F` (K` x)) -> (H` x) = (K` x)))
21	14, 16, 18, 20	syl12anc 1098	. . . . 5 |- (((F:B-1-1->C /\ H:A-->B /\ K:A-->B) /\ x e. A) -> ((F` (H` x)) = (F` (K` x)) -> (H` x) = (K` x)))
22	13, 21	sylbid 220	. . . 4 |- (((F:B-1-1->C /\ H:A-->B /\ K:A-->B) /\ x e. A) -> (((F o. H)` x) = ((F o. K)` x) -> (H` x) = (K` x)))
23	22	ralimdvaa 2171	. . 3 |- ((F:B-1-1->C /\ H:A-->B /\ K:A-->B) -> (A.x e. A ((F o. H)` x) = ((F o. K)` x) -> A.x e. A (H` x) = (K` x)))
24	23	anim2d 620	. 2 |- ((F:B-1-1->C /\ H:A-->B /\ K:A-->B) -> ((A = A /\ A.x e. A ((F o. H)` x) = ((F o. K)` x)) -> (A = A /\ A.x e. A (H` x) = (K` x))))
25		ffn 4562	. . . . . 6 |- (F:B-->C -> F Fn B)
26		fnfco 4581	. . . . . . 7 |- ((F Fn B /\ H:A-->B) -> (F o. H) Fn A)
27	26	ex 402	. . . . . 6 |- (F Fn B -> (H:A-->B -> (F o. H) Fn A))
28	1, 25, 27	3syl 24	. . . . 5 |- (F:B-1-1->C -> (H:A-->B -> (F o. H) Fn A))
29	28	imp 377	. . . 4 |- ((F:B-1-1->C /\ H:A-->B) -> (F o. H) Fn A)
30	29	3adant3 896	. . 3 |- ((F:B-1-1->C /\ H:A-->B /\ K:A-->B) -> (F o. H) Fn A)
31		fnfco 4581	. . . . . . 7 |- ((F Fn B /\ K:A-->B) -> (F o. K) Fn A)
32	31	ex 402	. . . . . 6 |- (F Fn B -> (K:A-->B -> (F o. K) Fn A))
33	1, 25, 32	3syl 24	. . . . 5 |- (F:B-1-1->C -> (K:A-->B -> (F o. K) Fn A))
34	33	imp 377	. . . 4 |- ((F:B-1-1->C /\ K:A-->B) -> (F o. K) Fn A)
35	34	3adant2 895	. . 3 |- ((F:B-1-1->C /\ H:A-->B /\ K:A-->B) -> (F o. K) Fn A)
36		eqfnfv 4766	. . 3 |- (((F o. H) Fn A /\ (F o. K) Fn A) -> ((F o. H) = (F o. K) <-> (A = A /\ A.x e. A ((F o. H)` x) = ((F o. K)` x))))
37	30, 35, 36	syl11anc 524	. 2 |- ((F:B-1-1->C /\ H:A-->B /\ K:A-->B) -> ((F o. H) = (F o. K) <-> (A = A /\ A.x e. A ((F o. H)` x) = ((F o. K)` x))))
38		ffn 4562	. . . . 5 |- (H:A-->B -> H Fn A)
39		ffn 4562	. . . . 5 |- (K:A-->B -> K Fn A)
40	38, 39	anim12i 360	. . . 4 |- ((H:A-->B /\ K:A-->B) -> (H Fn A /\ K Fn A))
41	40	3adant1 894	. . 3 |- ((F:B-1-1->C /\ H:A-->B /\ K:A-->B) -> (H Fn A /\ K Fn A))
42		eqfnfv 4766	. . 3 |- ((H Fn A /\ K Fn A) -> (H = K <-> (A = A /\ A.x e. A (H` x) = (K` x))))
43	41, 42	syl 12	. 2 |- ((F:B-1-1->C /\ H:A-->B /\ K:A-->B) -> (H = K <-> (A = A /\ A.x e. A (H` x) = (K` x))))
44	24, 37, 43	3imtr4d 602	1 |- ((F:B-1-1->C /\ H:A-->B /\ K:A-->B) -> ((F o. H) = (F o. K) -> H = K))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  A.wral 2105   o. ccom 3990  Fun wfun 3992   Fn wfn 3993  -->wf 3994  -1-1->wf1 3995  ` cfv 3998

This theorem is referenced by:  injsurinj 14487

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-3an 860  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