Theorem ismonb2 15161

Description: A monomorphism is a left-cancelable morphism.

Hypotheses

Ref	Expression
ismonb2.1	|- M = dom (dom` T)
ismonb2.2	|- D = (dom` T)
ismonb2.3	|- C = (cod` T)
ismonb2.4	|- R = (o` T)

Assertion

Ref	Expression
ismonb2	|- ((T e. Cat /\ (F e. M /\ G e. M /\ J e. M)) -> (F e. ( Monic ` T) -> (((D` G) = (D` J) /\ (C` G) = (D` F) /\ (C` J) = (D` F)) -> ((FRG) = (FRJ) -> G = J))))

Proof of Theorem ismonb2

Step	Hyp	Ref	Expression
1		ismonb2.1	. . . 4 |- M = dom (dom` T)
2		ismonb2.2	. . . 4 |- D = (dom` T)
3		ismonb2.3	. . . 4 |- C = (cod` T)
4		ismonb2.4	. . . 4 |- R = (o` T)
5	1, 2, 3, 4	ismonb1 15160	. . 3 |- ((T e. Cat /\ F e. M) -> (F e. ( Monic ` T) <-> A.g e. M A.j e. M (((D` g) = (D` j) /\ (C` g) = (D` F) /\ (C` j) = (D` F)) -> ((FRg) = (FRj) -> g = j))))
6	5	3ad2antr1 1041	. 2 |- ((T e. Cat /\ (F e. M /\ G e. M /\ J e. M)) -> (F e. ( Monic ` T) <-> A.g e. M A.j e. M (((D` g) = (D` j) /\ (C` g) = (D` F) /\ (C` j) = (D` F)) -> ((FRg) = (FRj) -> g = j))))
7		3simpc 874	. . . 4 |- ((F e. M /\ G e. M /\ J e. M) -> (G e. M /\ J e. M))
8	7	adantl 424	. . 3 |- ((T e. Cat /\ (F e. M /\ G e. M /\ J e. M)) -> (G e. M /\ J e. M))
9		fveq2 4681	. . . . . . 7 |- (g = G -> (D` g) = (D` G))
10	9	eqeq1d 1892	. . . . . 6 |- (g = G -> ((D` g) = (D` j) <-> (D` G) = (D` j)))
11		fveq2 4681	. . . . . . 7 |- (g = G -> (C` g) = (C` G))
12	11	eqeq1d 1892	. . . . . 6 |- (g = G -> ((C` g) = (D` F) <-> (C` G) = (D` F)))
13	10, 12	3anbi12d 1169	. . . . 5 |- (g = G -> (((D` g) = (D` j) /\ (C` g) = (D` F) /\ (C` j) = (D` F)) <-> ((D` G) = (D` j) /\ (C` G) = (D` F) /\ (C` j) = (D` F))))
14		opreq2 4890	. . . . . . 7 |- (g = G -> (FRg) = (FRG))
15	14	eqeq1d 1892	. . . . . 6 |- (g = G -> ((FRg) = (FRj) <-> (FRG) = (FRj)))
16		eqeq1 1890	. . . . . 6 |- (g = G -> (g = j <-> G = j))
17	15, 16	imbi12d 688	. . . . 5 |- (g = G -> (((FRg) = (FRj) -> g = j) <-> ((FRG) = (FRj) -> G = j)))
18	13, 17	imbi12d 688	. . . 4 |- (g = G -> ((((D` g) = (D` j) /\ (C` g) = (D` F) /\ (C` j) = (D` F)) -> ((FRg) = (FRj) -> g = j)) <-> (((D` G) = (D` j) /\ (C` G) = (D` F) /\ (C` j) = (D` F)) -> ((FRG) = (FRj) -> G = j))))
19		fveq2 4681	. . . . . . 7 |- (j = J -> (D` j) = (D` J))
20	19	eqeq2d 1895	. . . . . 6 |- (j = J -> ((D` G) = (D` j) <-> (D` G) = (D` J)))
21		fveq2 4681	. . . . . . 7 |- (j = J -> (C` j) = (C` J))
22	21	eqeq1d 1892	. . . . . 6 |- (j = J -> ((C` j) = (D` F) <-> (C` J) = (D` F)))
23	20, 22	3anbi13d 1170	. . . . 5 |- (j = J -> (((D` G) = (D` j) /\ (C` G) = (D` F) /\ (C` j) = (D` F)) <-> ((D` G) = (D` J) /\ (C` G) = (D` F) /\ (C` J) = (D` F))))
24		opreq2 4890	. . . . . . 7 |- (j = J -> (FRj) = (FRJ))
25	24	eqeq2d 1895	. . . . . 6 |- (j = J -> ((FRG) = (FRj) <-> (FRG) = (FRJ)))
26		eqeq2 1893	. . . . . 6 |- (j = J -> (G = j <-> G = J))
27	25, 26	imbi12d 688	. . . . 5 |- (j = J -> (((FRG) = (FRj) -> G = j) <-> ((FRG) = (FRJ) -> G = J)))
28	23, 27	imbi12d 688	. . . 4 |- (j = J -> ((((D` G) = (D` j) /\ (C` G) = (D` F) /\ (C` j) = (D` F)) -> ((FRG) = (FRj) -> G = j)) <-> (((D` G) = (D` J) /\ (C` G) = (D` F) /\ (C` J) = (D` F)) -> ((FRG) = (FRJ) -> G = J))))
29	18, 28	rcla42v 2384	. . 3 |- ((G e. M /\ J e. M) -> (A.g e. M A.j e. M (((D` g) = (D` j) /\ (C` g) = (D` F) /\ (C` j) = (D` F)) -> ((FRg) = (FRj) -> g = j)) -> (((D` G) = (D` J) /\ (C` G) = (D` F) /\ (C` J) = (D` F)) -> ((FRG) = (FRJ) -> G = J))))
30	8, 29	syl 12	. 2 |- ((T e. Cat /\ (F e. M /\ G e. M /\ J e. M)) -> (A.g e. M A.j e. M (((D` g) = (D` j) /\ (C` g) = (D` F) /\ (C` j) = (D` F)) -> ((FRg) = (FRj) -> g = j)) -> (((D` G) = (D` J) /\ (C` G) = (D` F) /\ (C` J) = (D` F)) -> ((FRG) = (FRJ) -> G = J))))
31	6, 30	sylbid 220	1 |- ((T e. Cat /\ (F e. M /\ G e. M /\ J e. M)) -> (F e. ( Monic ` T) -> (((D` G) = (D` J) /\ (C` G) = (D` F) /\ (C` J) = (D` F)) -> ((FRG) = (FRJ) -> G = J))))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  A.wral 2105  dom cdm 3986  ` cfv 3998  (class class class)co 4884  domcdom_ 15059  codccod_ 15060  oco_ 15062   Cat ccat 15099   Monic cmon 15153

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-fv 4014  df-opr 4886  df-mon 15155

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