Theorem domcmpd 15093

Description: When (G(o` T)F) is defined its domain is the domain of F.

Hypotheses

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

Assertion

Ref	Expression
domcmpd	|- ((T e. Ded /\ F e. M /\ G e. M) -> ((D` G) = (C` F) -> (D` (GRF)) = (D` F)))

Proof of Theorem domcmpd

Step	Hyp	Ref	Expression
1		ded.2	. . . 4 |- D = (dom` T)
2		ded.3	. . . 4 |- C = (cod` T)
3		eqid 1884	. . . 4 |- (id` T) = (id` T)
4		ded.4	. . . 4 |- R = (o` T)
5		ded.1	. . . 4 |- M = dom D
6		eqid 1884	. . . 4 |- dom (id` T) = dom (id` T)
7	1, 2, 3, 4, 5, 6	dedi 15084	. . 3 |- (T e. Ded -> ((<.<.D, C>., <.(id` T), R>.>. e. Alg /\ A.x e. dom (id` T)((D` ((id` T)` x)) = x /\ (C` ((id` T)` x)) = x) /\ A.f e. M A.g e. M (<.g, f>. e. dom R <-> (D` g) = (C` f))) /\ (A.f e. M A.g e. M ((D` g) = (C` f) -> (D` (gRf)) = (D` f)) /\ A.f e. M A.g e. M ((D` g) = (C` f) -> (C` (gRf)) = (C` g)))))
8		fveq2 4681	. . . . . . . 8 |- (f = F -> (C` f) = (C` F))
9	8	eqeq2d 1895	. . . . . . 7 |- (f = F -> ((D` g) = (C` f) <-> (D` g) = (C` F)))
10		opreq2 4890	. . . . . . . . 9 |- (f = F -> (gRf) = (gRF))
11	10	fveq2d 4685	. . . . . . . 8 |- (f = F -> (D` (gRf)) = (D` (gRF)))
12		fveq2 4681	. . . . . . . 8 |- (f = F -> (D` f) = (D` F))
13	11, 12	eqeq12d 1899	. . . . . . 7 |- (f = F -> ((D` (gRf)) = (D` f) <-> (D` (gRF)) = (D` F)))
14	9, 13	imbi12d 688	. . . . . 6 |- (f = F -> (((D` g) = (C` f) -> (D` (gRf)) = (D` f)) <-> ((D` g) = (C` F) -> (D` (gRF)) = (D` F))))
15		fveq2 4681	. . . . . . . 8 |- (g = G -> (D` g) = (D` G))
16	15	eqeq1d 1892	. . . . . . 7 |- (g = G -> ((D` g) = (C` F) <-> (D` G) = (C` F)))
17		opreq1 4889	. . . . . . . . 9 |- (g = G -> (gRF) = (GRF))
18	17	fveq2d 4685	. . . . . . . 8 |- (g = G -> (D` (gRF)) = (D` (GRF)))
19	18	eqeq1d 1892	. . . . . . 7 |- (g = G -> ((D` (gRF)) = (D` F) <-> (D` (GRF)) = (D` F)))
20	16, 19	imbi12d 688	. . . . . 6 |- (g = G -> (((D` g) = (C` F) -> (D` (gRF)) = (D` F)) <-> ((D` G) = (C` F) -> (D` (GRF)) = (D` F))))
21	14, 20	rcla42v 2384	. . . . 5 |- ((F e. M /\ G e. M) -> (A.f e. M A.g e. M ((D` g) = (C` f) -> (D` (gRf)) = (D` f)) -> ((D` G) = (C` F) -> (D` (GRF)) = (D` F))))
22	21	com12 14	. . . 4 |- (A.f e. M A.g e. M ((D` g) = (C` f) -> (D` (gRf)) = (D` f)) -> ((F e. M /\ G e. M) -> ((D` G) = (C` F) -> (D` (GRF)) = (D` F))))
23	22	ad2antrl 442	. . 3 |- (((<.<.D, C>., <.(id` T), R>.>. e. Alg /\ A.x e. dom (id` T)((D` ((id` T)` x)) = x /\ (C` ((id` T)` x)) = x) /\ A.f e. M A.g e. M (<.g, f>. e. dom R <-> (D` g) = (C` f))) /\ (A.f e. M A.g e. M ((D` g) = (C` f) -> (D` (gRf)) = (D` f)) /\ A.f e. M A.g e. M ((D` g) = (C` f) -> (C` (gRf)) = (C` g)))) -> ((F e. M /\ G e. M) -> ((D` G) = (C` F) -> (D` (GRF)) = (D` F))))
24	7, 23	syl 12	. 2 |- (T e. Ded -> ((F e. M /\ G e. M) -> ((D` G) = (C` F) -> (D` (GRF)) = (D` F))))
25	24	3impib 1065	1 |- ((T e. Ded /\ F e. M /\ G e. M) -> ((D` G) = (C` F) -> (D` (GRF)) = (D` F)))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  A.wral 2105  <.cop 3046  dom cdm 3986  ` cfv 3998  (class class class)co 4884   Alg calg 15058  domcdom_ 15059  codccod_ 15060  idcid_ 15061  oco_ 15062   Ded cded 15081

This theorem is referenced by:  domcmpc 15118  dualded 15132  homgrf 15151

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-ded 15082

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