Theorem cmpidb 15122

Description: The 11th "axiom" of a category: (J` A) is a right neutral element.

Hypotheses

Ref	Expression
cmpidb.1	|- M = dom D
cmpidb.2	|- D = (dom` T)
cmpidb.3	|- O = dom J
cmpidb.4	|- J = (id` T)
cmpidb.5	|- R = (o` T)

Assertion

Ref	Expression
cmpidb	|- ((T e. Cat /\ A e. O /\ F e. M) -> ((D` F) = A -> (FR(J` A)) = F))

Proof of Theorem cmpidb

Step	Hyp	Ref	Expression
1		cmpidb.2	. . . 4 |- D = (dom` T)
2		eqid 1884	. . . 4 |- (cod` T) = (cod` T)
3		cmpidb.4	. . . 4 |- J = (id` T)
4		cmpidb.5	. . . 4 |- R = (o` T)
5		cmpidb.1	. . . 4 |- M = dom D
6		cmpidb.3	. . . 4 |- O = dom J
7	1, 2, 3, 4, 5, 6	cati 15102	. . 3 |- (T e. Cat -> ((<.<.D, (cod` T)>., <.J, R>.>. e. Ded /\ A.f e. M A.x e. M A.y e. M (((D` y) = ((cod` T)` x) /\ (D` x) = ((cod` T)` f)) -> (yR(xRf)) = ((yRx)Rf))) /\ (A.a e. O A.f e. M (((cod` T)` f) = a -> ((J` a)Rf) = f) /\ A.a e. O A.f e. M ((D` f) = a -> (fR(J` a)) = f))))
8		eqeq2 1893	. . . . . . 7 |- (a = A -> ((D` f) = a <-> (D` f) = A))
9		fveq2 4681	. . . . . . . . 9 |- (a = A -> (J` a) = (J` A))
10	9	opreq2d 4898	. . . . . . . 8 |- (a = A -> (fR(J` a)) = (fR(J` A)))
11	10	eqeq1d 1892	. . . . . . 7 |- (a = A -> ((fR(J` a)) = f <-> (fR(J` A)) = f))
12	8, 11	imbi12d 688	. . . . . 6 |- (a = A -> (((D` f) = a -> (fR(J` a)) = f) <-> ((D` f) = A -> (fR(J` A)) = f)))
13		fveq2 4681	. . . . . . . 8 |- (f = F -> (D` f) = (D` F))
14	13	eqeq1d 1892	. . . . . . 7 |- (f = F -> ((D` f) = A <-> (D` F) = A))
15		opreq1 4889	. . . . . . . 8 |- (f = F -> (fR(J` A)) = (FR(J` A)))
16		id 73	. . . . . . . 8 |- (f = F -> f = F)
17	15, 16	eqeq12d 1899	. . . . . . 7 |- (f = F -> ((fR(J` A)) = f <-> (FR(J` A)) = F))
18	14, 17	imbi12d 688	. . . . . 6 |- (f = F -> (((D` f) = A -> (fR(J` A)) = f) <-> ((D` F) = A -> (FR(J` A)) = F)))
19	12, 18	rcla42v 2384	. . . . 5 |- ((A e. O /\ F e. M) -> (A.a e. O A.f e. M ((D` f) = a -> (fR(J` a)) = f) -> ((D` F) = A -> (FR(J` A)) = F)))
20	19	com12 14	. . . 4 |- (A.a e. O A.f e. M ((D` f) = a -> (fR(J` a)) = f) -> ((A e. O /\ F e. M) -> ((D` F) = A -> (FR(J` A)) = F)))
21	20	ad2antll 443	. . 3 |- (((<.<.D, (cod` T)>., <.J, R>.>. e. Ded /\ A.f e. M A.x e. M A.y e. M (((D` y) = ((cod` T)` x) /\ (D` x) = ((cod` T)` f)) -> (yR(xRf)) = ((yRx)Rf))) /\ (A.a e. O A.f e. M (((cod` T)` f) = a -> ((J` a)Rf) = f) /\ A.a e. O A.f e. M ((D` f) = a -> (fR(J` a)) = f))) -> ((A e. O /\ F e. M) -> ((D` F) = A -> (FR(J` A)) = F)))
22	7, 21	syl 12	. 2 |- (T e. Cat -> ((A e. O /\ F e. M) -> ((D` F) = A -> (FR(J` A)) = F)))
23	22	3impib 1065	1 |- ((T e. Cat /\ A e. O /\ F e. M) -> ((D` F) = A -> (FR(J` A)) = F))

Syntax hints:   -> wi 3   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  A.wral 2105  <.cop 3046  dom cdm 3986  ` cfv 3998  (class class class)co 4884  domcdom_ 15059  codccod_ 15060  idcid_ 15061  oco_ 15062   Ded cded 15081   Cat ccat 15099

This theorem is referenced by:  dualcat2 15133  cmphmib 15148

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-fo 4012  df-fv 4014  df-opr 4886  df-1st 5020  df-2nd 5021  df-doma 15064  df-coda 15065  df-ida 15066  df-cmpa 15067  df-cat 15100

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