Theorem codcmpd 10651

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

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

Proof of Theorem codcmpd

Step	Hyp	Ref	Expression
1		ded.2	. . . 4 |- D = (dom` T)
2		ded.3	. . . 4 |- C = (cod` T)
3		eqid 1478	. . . 4 |- (id` T) = (id` T)
4		ded.4	. . . 4 |- R = (o` T)
5		ded.1	. . . 4 |- M = dom D
6		eqid 1478	. . . 4 |- dom (id` T) = dom (id` T)
7	1, 2, 3, 4, 5, 6	dedi 10641	. . 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 3730	. . . . . . . 8 |- (f = F -> (C` f) = (C` F))
9	8	eqeq2d 1489	. . . . . . 7 |- (f = F -> ((D` g) = (C` f) <-> (D` g) = (C` F)))
10		opreq2 3975	. . . . . . . . 9 |- (f = F -> (gRf) = (gRF))
11	10	fveq2d 3734	. . . . . . . 8 |- (f = F -> (C` (gRf)) = (C` (gRF)))
12	11	eqeq1d 1486	. . . . . . 7 |- (f = F -> ((C` (gRf)) = (C` g) <-> (C` (gRF)) = (C` g)))
13	9, 12	imbi12d 628	. . . . . 6 |- (f = F -> (((D` g) = (C` f) -> (C` (gRf)) = (C` g)) <-> ((D` g) = (C` F) -> (C` (gRF)) = (C` g))))
14		fveq2 3730	. . . . . . . 8 |- (g = G -> (D` g) = (D` G))
15	14	eqeq1d 1486	. . . . . . 7 |- (g = G -> ((D` g) = (C` F) <-> (D` G) = (C` F)))
16		opreq1 3974	. . . . . . . . 9 |- (g = G -> (gRF) = (GRF))
17	16	fveq2d 3734	. . . . . . . 8 |- (g = G -> (C` (gRF)) = (C` (GRF)))
18		fveq2 3730	. . . . . . . 8 |- (g = G -> (C` g) = (C` G))
19	17, 18	eqeq12d 1492	. . . . . . 7 |- (g = G -> ((C` (gRF)) = (C` g) <-> (C` (GRF)) = (C` G)))
20	15, 19	imbi12d 628	. . . . . 6 |- (g = G -> (((D` g) = (C` F) -> (C` (gRF)) = (C` g)) <-> ((D` G) = (C` F) -> (C` (GRF)) = (C` G))))
21	13, 20	rcla42v 1883	. . . . 5 |- ((F e. M /\ G e. M) -> (A.f e. M A.g e. M ((D` g) = (C` f) -> (C` (gRf)) = (C` g)) -> ((D` G) = (C` F) -> (C` (GRF)) = (C` G))))
22	21	com12 11	. . . 4 |- (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) -> (C` (GRF)) = (C` G))))
23	22	ad2antll 409	. . 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) -> (C` (GRF)) = (C` G))))
24	7, 23	syl 10	. 2 |- (T e. Ded -> ((F e. M /\ G e. M) -> ((D` G) = (C` F) -> (C` (GRF)) = (C` G))))
25	24	3impib 833	1 |- ((T e. Ded /\ F e. M /\ G e. M) -> ((D` G) = (C` F) -> (C` (GRF)) = (C` G)))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   /\ w3a 777   = wceq 958   e. wcel 960  A.wral 1648  <.cop 2415  dom cdm 3176  ` cfv 3188  (class class class)co 3969  Algcalg 10614  domcdom_ 10615  codccod_ 10616  idcid_ 10617  oco_ 10618  Dedcded 10638

This theorem is referenced by:  codcmpc 10676  homgrf 10701

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-nul 2715  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-fn 3199  df-f 3200  df-fo 3202  df-fv 3204  df-opr 3971  df-1st 4085  df-2nd 4086  df-doma 10620  df-coda 10621  df-ida 10622  df-cmpa 10623  df-ded 10639

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