Theorem cinvlem3 15178

Description: The set of the inverses of the morphism F.

Hypotheses

Ref	Expression
cinvlem3.1	|- M = dom (dom` T)
cinvlem3.2	|- D = (dom` T)
cinvlem3.3	|- C = (cod` T)
cinvlem3.4	|- R = (o` T)
cinvlem3.5	|- J = (id` T)
cinvlem3.6	|- T e. Cat
cinvlem3.7	|- F e. M

Assertion

Ref	Expression
cinvlem3	|- (G e. (( cinv ` T)` F) <-> (G e. M /\ (FRG) = (J` (C` F)) /\ (GRF) = (J` (D` F))))

Proof of Theorem cinvlem3

Step	Hyp	Ref	Expression
1		opreq2 4890	. . . . 5 |- (g = G -> (FRg) = (FRG))
2	1	eqeq1d 1892	. . . 4 |- (g = G -> ((FRg) = (J` (C` F)) <-> (FRG) = (J` (C` F))))
3		opreq1 4889	. . . . 5 |- (g = G -> (gRF) = (GRF))
4	3	eqeq1d 1892	. . . 4 |- (g = G -> ((gRF) = (J` (D` F)) <-> (GRF) = (J` (D` F))))
5	2, 4	anbi12d 690	. . 3 |- (g = G -> (((FRg) = (J` (C` F)) /\ (gRF) = (J` (D` F))) <-> ((FRG) = (J` (C` F)) /\ (GRF) = (J` (D` F)))))
6	5	elrab 2414	. 2 |- (G e. {g e. M | ((FRg) = (J` (C` F)) /\ (gRF) = (J` (D` F)))} <-> (G e. M /\ ((FRG) = (J` (C` F)) /\ (GRF) = (J` (D` F)))))
7		cinvlem3.7	. . . 4 |- F e. M
8		cinvlem3.1	. . . . 5 |- M = dom (dom` T)
9		cinvlem3.2	. . . . 5 |- D = (dom` T)
10		cinvlem3.3	. . . . 5 |- C = (cod` T)
11		cinvlem3.4	. . . . 5 |- R = (o` T)
12		cinvlem3.5	. . . . 5 |- J = (id` T)
13		cinvlem3.6	. . . . 5 |- T e. Cat
14	8, 9, 10, 11, 12, 13	cinvlem2 15177	. . . 4 |- (F e. M -> (( cinv ` T)` F) = {g e. M | ((FRg) = (J` (C` F)) /\ (gRF) = (J` (D` F)))})
15	7, 14	ax-mp 7	. . 3 |- (( cinv ` T)` F) = {g e. M | ((FRg) = (J` (C` F)) /\ (gRF) = (J` (D` F)))}
16	15	eleq2i 1961	. 2 |- (G e. (( cinv ` T)` F) <-> G e. {g e. M | ((FRg) = (J` (C` F)) /\ (gRF) = (J` (D` F)))})
17		3anass 862	. 2 |- ((G e. M /\ (FRG) = (J` (C` F)) /\ (GRF) = (J` (D` F))) <-> (G e. M /\ ((FRG) = (J` (C` F)) /\ (GRF) = (J` (D` F)))))
18	6, 16, 17	3bitr4i 200	1 |- (G e. (( cinv ` T)` F) <-> (G e. M /\ (FRG) = (J` (C` F)) /\ (GRF) = (J` (D` F))))

Syntax hints:   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  {crab 2108  dom cdm 3986  ` cfv 3998  (class class class)co 4884  domcdom_ 15059  codccod_ 15060  idcid_ 15061  oco_ 15062   Cat ccat 15099   cinv ccinv 15174

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-rep 3428  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-rex 2110  df-rab 2112  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-fv 4014  df-opr 4886  df-cinv 15175

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