Theorem njtlc 15360

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 4736	. . . . . . . . . 10 |- (F:B-1-1->C -> F:B-->C)
2		ffun 4700	. . . . . . . . . 10 |- (F:B-->C -> Fun F)
3	1, 2	syl 13	. . . . . . . . 9 |- (F:B-1-1->C -> Fun F)
4	3	3ad2ant1 1169	. . . . . . . 8 |- ((F:B-1-1->C /\ H:A-->B /\ K:A-->B) -> Fun F)
5	4	adantr 543	. . . . . . 7 |- (((F:B-1-1->C /\ H:A-->B /\ K:A-->B) /\ x e. A) -> Fun F)
6		simpl2 1152	. . . . . . 7 |- (((F:B-1-1->C /\ H:A-->B /\ K:A-->B) /\ x e. A) -> H:A-->B)
7		simpr 538	. . . . . . 7 |- (((F:B-1-1->C /\ H:A-->B /\ K:A-->B) /\ x e. A) -> x e. A)
8		fvco3 4863	. . . . . . 7 |- ((Fun F /\ H:A-->B /\ x e. A) -> ((F o. H)` x) = (F` (H` x)))
9	5, 6, 7, 8	syl111anc 1377	. . . . . 6 |- (((F:B-1-1->C /\ H:A-->B /\ K:A-->B) /\ x e. A) -> ((F o. H)` x) = (F` (H` x)))
10		simpl3 1153	. . . . . . 7 |- (((F:B-1-1->C /\ H:A-->B /\ K:A-->B) /\ x e. A) -> K:A-->B)
11		fvco3 4863	. . . . . . 7 |- ((Fun F /\ K:A-->B /\ x e. A) -> ((F o. K)` x) = (F` (K` x)))
12	5, 10, 7, 11	syl111anc 1377	. . . . . 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 2184	. . . . 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 1151	. . . . . 6 |- (((F:B-1-1->C /\ H:A-->B /\ K:A-->B) /\ x e. A) -> F:B-1-1->C)
15		ffvelrn 4916	. . . . . . 7 |- ((H:A-->B /\ x e. A) -> (H` x) e. B)
16	15	3ad2antl2 1317	. . . . . 6 |- (((F:B-1-1->C /\ H:A-->B /\ K:A-->B) /\ x e. A) -> (H` x) e. B)
17		ffvelrn 4916	. . . . . . 7 |- ((K:A-->B /\ x e. A) -> (K` x) e. B)
18	17	3ad2antl3 1318	. . . . . 6 |- (((F:B-1-1->C /\ H:A-->B /\ K:A-->B) /\ x e. A) -> (K` x) e. B)
19		f1fveq 4986	. . . . . . 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 244	. . . . . 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 1375	. . . . 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 267	. . . 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	ralimdva 2451	. . 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 631	. 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 4698	. . . . . 6 |- (F:B-->C -> F Fn B)
26		fnfco 4713	. . . . . . 7 |- ((F Fn B /\ H:A-->B) -> (F o. H) Fn A)
27	26	ex 494	. . . . . 6 |- (F Fn B -> (H:A-->B -> (F o. H) Fn A))
28	1, 25, 27	3syl 38	. . . . 5 |- (F:B-1-1->C -> (H:A-->B -> (F o. H) Fn A))
29	28	imp 489	. . . 4 |- ((F:B-1-1->C /\ H:A-->B) -> (F o. H) Fn A)
30	29	3adant3 1168	. . 3 |- ((F:B-1-1->C /\ H:A-->B /\ K:A-->B) -> (F o. H) Fn A)
31		fnfco 4713	. . . . . . 7 |- ((F Fn B /\ K:A-->B) -> (F o. K) Fn A)
32	31	ex 494	. . . . . 6 |- (F Fn B -> (K:A-->B -> (F o. K) Fn A))
33	1, 25, 32	3syl 38	. . . . 5 |- (F:B-1-1->C -> (K:A-->B -> (F o. K) Fn A))
34	33	imp 489	. . . 4 |- ((F:B-1-1->C /\ K:A-->B) -> (F o. K) Fn A)
35	34	3adant2 1167	. . 3 |- ((F:B-1-1->C /\ H:A-->B /\ K:A-->B) -> (F o. K) Fn A)
36		eqfnfv 4892	. . 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 755	. 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 4698	. . . . 5 |- (H:A-->B -> H Fn A)
39		ffn 4698	. . . . 5 |- (K:A-->B -> K Fn A)
40	38, 39	anim12i 632	. . . 4 |- ((H:A-->B /\ K:A-->B) -> (H Fn A /\ K Fn A))
41	40	3adant1 1166	. . 3 |- ((F:B-1-1->C /\ H:A-->B /\ K:A-->B) -> (H Fn A /\ K Fn A))
42		eqfnfv 4892	. . 3 |- ((H Fn A /\ K Fn A) -> (H = K <-> (A = A /\ A.x e. A (H` x) = (K` x))))
43	41, 42	syl 13	. 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 331	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 231   /\ wa 433   /\ w3a 1130   = wceq 1615   e. wcel 1617  A.wral 2385   o. ccom 4155  Fun wfun 4157   Fn wfn 4158  -->wf 4159  -1-1->wf1 4160  ` cfv 4163

This theorem is referenced by:  injsurinj 15487

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1621  ax-gen 1622  ax-8 1623  ax-9 1624  ax-10 1625  ax-11 1626  ax-12 1627  ax-13 1628  ax-14 1629  ax-17 1634  ax-4 1637  ax-5o 1639  ax-6o 1642  ax-9o 1792  ax-10o 1810  ax-16 1883  ax-11o 1893  ax-ext 2152  ax-sep 3638  ax-nul 3645  ax-pow 3681  ax-pr 3719  ax-un 3961

This theorem depends on definitions:  df-bi 232  df-or 434  df-an 435  df-3an 1132  df-ex 1645  df-sb 1845  df-eu 2070  df-mo 2071  df-clab 2158  df-cleq 2163  df-clel 2166  df-ne 2297  df-ral 2389  df-rex 2390  df-v 2571  df-dif 2862  df-un 2864  df-in 2866  df-ss 2868  df-nul 3115  df-pw 3261  df-sn 3274  df-pr 3275  df-op 3278  df-uni 3399  df-br 3540  df-opab 3598  df-id 3779  df-xp 4165  df-rel 4166  df-cnv 4167  df-co 4168  df-dm 4169  df-rn 4170  df-res 4171  df-ima 4172  df-fun 4173  df-fn 4174  df-f 4175  df-f1 4176  df-fv 4179

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