Theorem isepib2 10721

Description: If F "is an epimorphism" is cancelable when it is the right operand of a composition.

Hypotheses

Ref	Expression
isepib2.1	|- M = dom (dom` T)
isepib2.2	|- D = (dom` T)
isepib2.3	|- C = (cod` T)
isepib2.4	|- R = (o` T)

Assertion

Ref	Expression
isepib2	|- ((T e. Cat /\ (F e. M /\ G e. M /\ J e. M)) -> (F e. (Epi` T) -> (((C` G) = (C` J) /\ (D` G) = (C` F) /\ (D` J) = (C` F)) -> ((GRF) = (JRF) -> G = J))))

Proof of Theorem isepib2

Step	Hyp	Ref	Expression
1		isepib2.1	. . . 4 |- M = dom (dom` T)
2		isepib2.2	. . . 4 |- D = (dom` T)
3		isepib2.3	. . . 4 |- C = (cod` T)
4		isepib2.4	. . . 4 |- R = (o` T)
5	1, 2, 3, 4	isepib1 10720	. . 3 |- ((T e. Cat /\ F e. M) -> (F e. (Epi` T) <-> A.g e. M A.j e. M (((C` g) = (C` j) /\ (D` g) = (C` F) /\ (D` j) = (C` F)) -> ((gRF) = (jRF) -> g = j))))
6	5	3ad2antr1 814	. 2 |- ((T e. Cat /\ (F e. M /\ G e. M /\ J e. M)) -> (F e. (Epi` T) <-> A.g e. M A.j e. M (((C` g) = (C` j) /\ (D` g) = (C` F) /\ (D` j) = (C` F)) -> ((gRF) = (jRF) -> g = j))))
7		3simpc 789	. . . 4 |- ((F e. M /\ G e. M /\ J e. M) -> (G e. M /\ J e. M))
8	7	adantl 390	. . 3 |- ((T e. Cat /\ (F e. M /\ G e. M /\ J e. M)) -> (G e. M /\ J e. M))
9		fveq2 3730	. . . . . . 7 |- (g = G -> (C` g) = (C` G))
10	9	eqeq1d 1486	. . . . . 6 |- (g = G -> ((C` g) = (C` j) <-> (C` G) = (C` j)))
11		fveq2 3730	. . . . . . 7 |- (g = G -> (D` g) = (D` G))
12	11	eqeq1d 1486	. . . . . 6 |- (g = G -> ((D` g) = (C` F) <-> (D` G) = (C` F)))
13	10, 12	3anbi12d 896	. . . . 5 |- (g = G -> (((C` g) = (C` j) /\ (D` g) = (C` F) /\ (D` j) = (C` F)) <-> ((C` G) = (C` j) /\ (D` G) = (C` F) /\ (D` j) = (C` F))))
14		opreq1 3974	. . . . . . 7 |- (g = G -> (gRF) = (GRF))
15	14	eqeq1d 1486	. . . . . 6 |- (g = G -> ((gRF) = (jRF) <-> (GRF) = (jRF)))
16		eqeq1 1484	. . . . . 6 |- (g = G -> (g = j <-> G = j))
17	15, 16	imbi12d 628	. . . . 5 |- (g = G -> (((gRF) = (jRF) -> g = j) <-> ((GRF) = (jRF) -> G = j)))
18	13, 17	imbi12d 628	. . . 4 |- (g = G -> ((((C` g) = (C` j) /\ (D` g) = (C` F) /\ (D` j) = (C` F)) -> ((gRF) = (jRF) -> g = j)) <-> (((C` G) = (C` j) /\ (D` G) = (C` F) /\ (D` j) = (C` F)) -> ((GRF) = (jRF) -> G = j))))
19		fveq2 3730	. . . . . . 7 |- (j = J -> (C` j) = (C` J))
20	19	eqeq2d 1489	. . . . . 6 |- (j = J -> ((C` G) = (C` j) <-> (C` G) = (C` J)))
21		fveq2 3730	. . . . . . 7 |- (j = J -> (D` j) = (D` J))
22	21	eqeq1d 1486	. . . . . 6 |- (j = J -> ((D` j) = (C` F) <-> (D` J) = (C` F)))
23	20, 22	3anbi13d 897	. . . . 5 |- (j = J -> (((C` G) = (C` j) /\ (D` G) = (C` F) /\ (D` j) = (C` F)) <-> ((C` G) = (C` J) /\ (D` G) = (C` F) /\ (D` J) = (C` F))))
24		opreq1 3974	. . . . . . 7 |- (j = J -> (jRF) = (JRF))
25	24	eqeq2d 1489	. . . . . 6 |- (j = J -> ((GRF) = (jRF) <-> (GRF) = (JRF)))
26		eqeq2 1487	. . . . . 6 |- (j = J -> (G = j <-> G = J))
27	25, 26	imbi12d 628	. . . . 5 |- (j = J -> (((GRF) = (jRF) -> G = j) <-> ((GRF) = (JRF) -> G = J)))
28	23, 27	imbi12d 628	. . . 4 |- (j = J -> ((((C` G) = (C` j) /\ (D` G) = (C` F) /\ (D` j) = (C` F)) -> ((GRF) = (jRF) -> G = j)) <-> (((C` G) = (C` J) /\ (D` G) = (C` F) /\ (D` J) = (C` F)) -> ((GRF) = (JRF) -> G = J))))
29	18, 28	rcla42v 1883	. . 3 |- ((G e. M /\ J e. M) -> (A.g e. M A.j e. M (((C` g) = (C` j) /\ (D` g) = (C` F) /\ (D` j) = (C` F)) -> ((gRF) = (jRF) -> g = j)) -> (((C` G) = (C` J) /\ (D` G) = (C` F) /\ (D` J) = (C` F)) -> ((GRF) = (JRF) -> G = J))))
30	8, 29	syl 10	. 2 |- ((T e. Cat /\ (F e. M /\ G e. M /\ J e. M)) -> (A.g e. M A.j e. M (((C` g) = (C` j) /\ (D` g) = (C` F) /\ (D` j) = (C` F)) -> ((gRF) = (jRF) -> g = j)) -> (((C` G) = (C` J) /\ (D` G) = (C` F) /\ (D` J) = (C` F)) -> ((GRF) = (JRF) -> G = J))))
31	6, 30	sylbid 203	1 |- ((T e. Cat /\ (F e. M /\ G e. M /\ J e. M)) -> (F e. (Epi` T) -> (((C` G) = (C` J) /\ (D` G) = (C` F) /\ (D` J) = (C` F)) -> ((GRF) = (JRF) -> G = J))))

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  Epicepi 10702

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-epi 10706

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