Theorem cmppfd 10678

Description: (G(o` T)F) is only defined when the domain of G is the codomain 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
cmppfd	|- ((T e. Ded /\ F e. M /\ G e. M) -> (<.G, F>. e. dom R <-> (D` G) = (C` F)))

Proof of Theorem cmppfd

Step	Hyp	Ref	Expression
1		ded.2	. . . 4 |- D = (dom` T)
2		ded.3	. . . 4 |- C = (cod` T)
3		eqid 1475	. . . 4 |- (id` T) = (id` T)
4		ded.4	. . . 4 |- R = (o` T)
5		ded.1	. . . 4 |- M = dom D
6		eqid 1475	. . . 4 |- dom (id` T) = dom (id` T)
7	1, 2, 3, 4, 5, 6	dedi 10670	. . 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		opeq2 2488	. . . . . . . . 9 |- (f = F -> <.g, f>. = <.g, F>.)
9	8	eleq1d 1540	. . . . . . . 8 |- (f = F -> (<.g, f>. e. dom R <-> <.g, F>. e. dom R))
10		fveq2 3724	. . . . . . . . 9 |- (f = F -> (C` f) = (C` F))
11	10	eqeq2d 1486	. . . . . . . 8 |- (f = F -> ((D` g) = (C` f) <-> (D` g) = (C` F)))
12	9, 11	bibi12d 629	. . . . . . 7 |- (f = F -> ((<.g, f>. e. dom R <-> (D` g) = (C` f)) <-> (<.g, F>. e. dom R <-> (D` g) = (C` F))))
13		opeq1 2487	. . . . . . . . 9 |- (g = G -> <.g, F>. = <.G, F>.)
14	13	eleq1d 1540	. . . . . . . 8 |- (g = G -> (<.g, F>. e. dom R <-> <.G, F>. e. dom R))
15		fveq2 3724	. . . . . . . . 9 |- (g = G -> (D` g) = (D` G))
16	15	eqeq1d 1483	. . . . . . . 8 |- (g = G -> ((D` g) = (C` F) <-> (D` G) = (C` F)))
17	14, 16	bibi12d 629	. . . . . . 7 |- (g = G -> ((<.g, F>. e. dom R <-> (D` g) = (C` F)) <-> (<.G, F>. e. dom R <-> (D` G) = (C` F))))
18	12, 17	rcla42v 1880	. . . . . 6 |- ((F e. M /\ G e. M) -> (A.f e. M A.g e. M (<.g, f>. e. dom R <-> (D` g) = (C` f)) -> (<.G, F>. e. dom R <-> (D` G) = (C` F))))
19	18	com12 11	. . . . 5 |- (A.f e. M A.g e. M (<.g, f>. e. dom R <-> (D` g) = (C` f)) -> ((F e. M /\ G e. M) -> (<.G, F>. e. dom R <-> (D` G) = (C` F))))
20	19	3ad2ant3 802	. . . 4 |- ((<.<.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))) -> ((F e. M /\ G e. M) -> (<.G, F>. e. dom R <-> (D` G) = (C` F))))
21	20	adantr 389	. . 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) -> (<.G, F>. e. dom R <-> (D` G) = (C` F))))
22	7, 21	syl 10	. 2 |- (T e. Ded -> ((F e. M /\ G e. M) -> (<.G, F>. e. dom R <-> (D` G) = (C` F))))
23	22	3impib 831	1 |- ((T e. Ded /\ F e. M /\ G e. M) -> (<.G, F>. e. dom R <-> (D` G) = (C` F)))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   /\ w3a 775   = wceq 956   e. wcel 958  A.wral 1645  <.cop 2411  dom cdm 3170  ` cfv 3182  (class class class)co 3963  Algcalg 10643  domcdom_ 10644  codccod_ 10645  idcid_ 10646  oco_ 10647  Dedcded 10667

This theorem is referenced by:  cmppfcd 10703

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-fo 3196  df-fv 3198  df-opr 3965  df-1st 4079  df-2nd 4080  df-doma 10649  df-coda 10650  df-ida 10651  df-cmpa 10652  df-ded 10668

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