Theorem mrdmcd 10722

Description: A morphism belongs to the homset between its domain and its codomain. JFM CAT₁ th. 22.

Hypotheses

Ref	Expression
mrdmcd.1	|- M = dom (dom` T)
mrdmcd.2	|- H = (hom` T)
mrdmcd.3	|- D = (dom` T)
mrdmcd.4	|- C = (cod` T)

Assertion

Ref	Expression
mrdmcd	|- (T e. Cat -> (F e. M -> F e. (H` <.(D` F), (C` F)>.)))

Proof of Theorem mrdmcd

Step	Hyp	Ref	Expression
1		catded 10697	. . . 4 |- (T e. Cat -> T e. Ded)
2		dedalg 10676	. . . 4 |- (T e. Ded -> T e. Alg)
3		mrdmcd.1	. . . . . . 7 |- M = dom (dom` T)
4		mrdmcd.3	. . . . . . . . 9 |- D = (dom` T)
5	4	eqcomi 1479	. . . . . . . 8 |- (dom` T) = D
6	5	dmeqi 3312	. . . . . . 7 |- dom (dom` T) = dom D
7	3, 6	eqtr 1495	. . . . . 6 |- M = dom D
8		eqid 1475	. . . . . 6 |- dom (id` T) = dom (id` T)
9		eqid 1475	. . . . . 6 |- (id` T) = (id` T)
10	7, 4, 8, 9	doma 10661	. . . . 5 |- (T e. Alg -> D:M-->dom (id` T))
11		ffvelrn 3814	. . . . . 6 |- ((D:M-->dom (id` T) /\ F e. M) -> (D` F) e. dom (id` T))
12	11	ex 373	. . . . 5 |- (D:M-->dom (id` T) -> (F e. M -> (D` F) e. dom (id` T)))
13	10, 12	syl 10	. . . 4 |- (T e. Alg -> (F e. M -> (D` F) e. dom (id` T)))
14	1, 2, 13	3syl 20	. . 3 |- (T e. Cat -> (F e. M -> (D` F) e. dom (id` T)))
15		mrdmcd.4	. . . . . 6 |- C = (cod` T)
16	7, 4, 8, 9, 15	coda 10662	. . . . 5 |- (T e. Alg -> C:M-->dom (id` T))
17		ffvelrn 3814	. . . . . 6 |- ((C:M-->dom (id` T) /\ F e. M) -> (C` F) e. dom (id` T))
18	17	ex 373	. . . . 5 |- (C:M-->dom (id` T) -> (F e. M -> (C` F) e. dom (id` T)))
19	16, 18	syl 10	. . . 4 |- (T e. Alg -> (F e. M -> (C` F) e. dom (id` T)))
20	1, 2, 19	3syl 20	. . 3 |- (T e. Cat -> (F e. M -> (C` F) e. dom (id` T)))
21	14, 20	jcad 600	. 2 |- (T e. Cat -> (F e. M -> ((D` F) e. dom (id` T) /\ (C` F) e. dom (id` T))))
22		eqid 1475	. . . . 5 |- (D` F) = (D` F)
23		eqid 1475	. . . . 5 |- (C` F) = (C` F)
24		mrdmcd.2	. . . . . . . . 9 |- H = (hom` T)
25	8, 3, 4, 15, 24	ishomd 10718	. . . . . . . 8 |- ((T e. Cat /\ (D` F) e. dom (id` T) /\ (C` F) e. dom (id` T)) -> (F e. (H` <.(D` F), (C` F)>.) <-> (F e. M /\ (D` F) = (D` F) /\ (C` F) = (C` F))))
26	25	biimprcd 156	. . . . . . 7 |- ((F e. M /\ (D` F) = (D` F) /\ (C` F) = (C` F)) -> ((T e. Cat /\ (D` F) e. dom (id` T) /\ (C` F) e. dom (id` T)) -> F e. (H` <.(D` F), (C` F)>.)))
27	26	3expib 836	. . . . . 6 |- (F e. M -> (((D` F) = (D` F) /\ (C` F) = (C` F)) -> ((T e. Cat /\ (D` F) e. dom (id` T) /\ (C` F) e. dom (id` T)) -> F e. (H` <.(D` F), (C` F)>.))))
28	27	com3l 34	. . . . 5 |- (((D` F) = (D` F) /\ (C` F) = (C` F)) -> ((T e. Cat /\ (D` F) e. dom (id` T) /\ (C` F) e. dom (id` T)) -> (F e. M -> F e. (H` <.(D` F), (C` F)>.))))
29	22, 23, 28	mp2an 697	. . . 4 |- ((T e. Cat /\ (D` F) e. dom (id` T) /\ (C` F) e. dom (id` T)) -> (F e. M -> F e. (H` <.(D` F), (C` F)>.)))
30	29	3expib 836	. . 3 |- (T e. Cat -> (((D` F) e. dom (id` T) /\ (C` F) e. dom (id` T)) -> (F e. M -> F e. (H` <.(D` F), (C` F)>.))))
31	30	com23 32	. 2 |- (T e. Cat -> (F e. M -> (((D` F) e. dom (id` T) /\ (C` F) e. dom (id` T)) -> F e. (H` <.(D` F), (C` F)>.))))
32	21, 31	mpdd 46	1 |- (T e. Cat -> (F e. M -> F e. (H` <.(D` F), (C` F)>.)))

Syntax hints:   -> wi 3   /\ wa 223   /\ w3a 775   = wceq 956   e. wcel 958  <.cop 2411  dom cdm 3170  -->wf 3178  ` cfv 3182  Algcalg 10643  domcdom_ 10644  codccod_ 10645  idcid_ 10646  Dedcded 10667  Catccat 10685  homchom 10713

This theorem is referenced by:  homib 10724

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-rep 2693  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-sbc 1942  df-csb 2002  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-if 2362  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-int 2534  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-f1 3195  df-fo 3196  df-f1o 3197  df-fv 3198  df-opr 3965  df-oprab 3966  df-1st 4079  df-2nd 4080  df-alg 10648  df-doma 10649  df-coda 10650  df-ida 10651  df-cmpa 10652  df-ded 10668  df-cat 10686  df-hom 10714

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