Theorem ismonb 15159

Description: The predicate "is a monomorphism".

Hypotheses

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

Assertion

Ref	Expression
ismonb	|- ((T e. Cat /\ F e. A) -> (F e. ( Monic ` T) <-> (F e. M /\ A.g e. M A.h e. M (((D` g) = (D` h) /\ (C` g) = (D` F) /\ (C` h) = (D` F)) -> ((FRg) = (FRh) -> g = h)))))

Distinct variable groups:   g,F,h   g,M,h   T,g,h

Proof of Theorem ismonb

Step	Hyp	Ref	Expression
1		ismonb.1	. . . . 5 |- M = dom (dom` T)
2		ismonb.2	. . . . 5 |- D = (dom` T)
3		ismonb.3	. . . . 5 |- C = (cod` T)
4		ismonb.4	. . . . 5 |- R = (o` T)
5	1, 2, 3, 4	ismona 15158	. . . 4 |- (T e. Cat -> ( Monic ` T) = {f | (f e. M /\ A.g e. M A.h e. M (((D` g) = (D` h) /\ (C` g) = (D` f) /\ (C` h) = (D` f)) -> ((fRg) = (fRh) -> g = h)))})
6	5	adantr 425	. . 3 |- ((T e. Cat /\ F e. A) -> ( Monic ` T) = {f | (f e. M /\ A.g e. M A.h e. M (((D` g) = (D` h) /\ (C` g) = (D` f) /\ (C` h) = (D` f)) -> ((fRg) = (fRh) -> g = h)))})
7	6	eleq2d 1964	. 2 |- ((T e. Cat /\ F e. A) -> (F e. ( Monic ` T) <-> F e. {f | (f e. M /\ A.g e. M A.h e. M (((D` g) = (D` h) /\ (C` g) = (D` f) /\ (C` h) = (D` f)) -> ((fRg) = (fRh) -> g = h)))}))
8		eleq1 1957	. . . . 5 |- (f = F -> (f e. M <-> F e. M))
9		fveq2 4681	. . . . . . . . 9 |- (f = F -> (D` f) = (D` F))
10	9	eqeq2d 1895	. . . . . . . 8 |- (f = F -> ((C` g) = (D` f) <-> (C` g) = (D` F)))
11	9	eqeq2d 1895	. . . . . . . 8 |- (f = F -> ((C` h) = (D` f) <-> (C` h) = (D` F)))
12	10, 11	3anbi23d 1171	. . . . . . 7 |- (f = F -> (((D` g) = (D` h) /\ (C` g) = (D` f) /\ (C` h) = (D` f)) <-> ((D` g) = (D` h) /\ (C` g) = (D` F) /\ (C` h) = (D` F))))
13		opreq1 4889	. . . . . . . . 9 |- (f = F -> (fRg) = (FRg))
14		opreq1 4889	. . . . . . . . 9 |- (f = F -> (fRh) = (FRh))
15	13, 14	eqeq12d 1899	. . . . . . . 8 |- (f = F -> ((fRg) = (fRh) <-> (FRg) = (FRh)))
16	15	imbi1d 675	. . . . . . 7 |- (f = F -> (((fRg) = (fRh) -> g = h) <-> ((FRg) = (FRh) -> g = h)))
17	12, 16	imbi12d 688	. . . . . 6 |- (f = F -> ((((D` g) = (D` h) /\ (C` g) = (D` f) /\ (C` h) = (D` f)) -> ((fRg) = (fRh) -> g = h)) <-> (((D` g) = (D` h) /\ (C` g) = (D` F) /\ (C` h) = (D` F)) -> ((FRg) = (FRh) -> g = h))))
18	17	2ralbidv 2140	. . . . 5 |- (f = F -> (A.g e. M A.h e. M (((D` g) = (D` h) /\ (C` g) = (D` f) /\ (C` h) = (D` f)) -> ((fRg) = (fRh) -> g = h)) <-> A.g e. M A.h e. M (((D` g) = (D` h) /\ (C` g) = (D` F) /\ (C` h) = (D` F)) -> ((FRg) = (FRh) -> g = h))))
19	8, 18	anbi12d 690	. . . 4 |- (f = F -> ((f e. M /\ A.g e. M A.h e. M (((D` g) = (D` h) /\ (C` g) = (D` f) /\ (C` h) = (D` f)) -> ((fRg) = (fRh) -> g = h))) <-> (F e. M /\ A.g e. M A.h e. M (((D` g) = (D` h) /\ (C` g) = (D` F) /\ (C` h) = (D` F)) -> ((FRg) = (FRh) -> g = h)))))
20	19	elabg 2405	. . 3 |- (F e. A -> (F e. {f | (f e. M /\ A.g e. M A.h e. M (((D` g) = (D` h) /\ (C` g) = (D` f) /\ (C` h) = (D` f)) -> ((fRg) = (fRh) -> g = h)))} <-> (F e. M /\ A.g e. M A.h e. M (((D` g) = (D` h) /\ (C` g) = (D` F) /\ (C` h) = (D` F)) -> ((FRg) = (FRh) -> g = h)))))
21	20	adantl 424	. 2 |- ((T e. Cat /\ F e. A) -> (F e. {f | (f e. M /\ A.g e. M A.h e. M (((D` g) = (D` h) /\ (C` g) = (D` f) /\ (C` h) = (D` f)) -> ((fRg) = (fRh) -> g = h)))} <-> (F e. M /\ A.g e. M A.h e. M (((D` g) = (D` h) /\ (C` g) = (D` F) /\ (C` h) = (D` F)) -> ((FRg) = (FRh) -> g = h)))))
22	7, 21	bitrd 587	1 |- ((T e. Cat /\ F e. A) -> (F e. ( Monic ` T) <-> (F e. M /\ A.g e. M A.h e. M (((D` g) = (D` h) /\ (C` g) = (D` F) /\ (C` h) = (D` F)) -> ((FRg) = (FRh) -> g = h)))))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  {cab 1871  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 is referenced by:  ismonb1 15160  ismonc 15163

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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