Theorem ismonb1 10710

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
ismonb1	|- ((T e. Cat /\ F e. M) -> (F e. (Monic` T) <-> 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 ismonb1

Step	Hyp	Ref	Expression
1		ismonb.1	. . 3 |- M = dom (dom` T)
2		ismonb.2	. . 3 |- D = (dom` T)
3		ismonb.3	. . 3 |- C = (cod` T)
4		ismonb.4	. . 3 |- R = (o` T)
5	1, 2, 3, 4	ismonb 10709	. 2 |- ((T e. Cat /\ F e. M) -> (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)))))
6		ibar 645	. . 3 |- (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)))))
7	6	adantl 390	. 2 |- ((T e. Cat /\ 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)))))
8	5, 7	bitr4d 533	1 |- ((T e. Cat /\ F e. M) -> (F e. (Monic` T) <-> 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 146   /\ wa 223   /\ w3a 777   = wceq 958   e. wcel 960  A.wral 1648  dom cdm 3176  ` cfv 3188  (class class class)co 3969  domcdom_ 10615  codccod_ 10616  oco_ 10618  Catccat 10656  Moniccmon 10703

This theorem is referenced by:  ismonb2 10711

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fv 3204  df-opr 3971  df-mon 10705

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