Theorem ishoma 10715

Description: Definition of (hom` T).

Hypotheses

Ref	Expression
ishoma.1	|- O = dom (id` T)
ishoma.2	|- M = dom (dom` T)
ishoma.3	|- D = (dom` T)
ishoma.4	|- C = (cod` T)

Assertion

Ref	Expression
ishoma	|- (T e. Cat -> (hom` T) = {<.<.a, b>., c>. | (a e. O /\ b e. O /\ c = {f | (f e. M /\ (D` f) = a /\ (C` f) = b)})})

Distinct variable groups:   C,c   D,c   M,c   T,a,b,c,f

Proof of Theorem ishoma

Step	Hyp	Ref	Expression
1		fveq2 3724	. . . . . . 7 |- (x = T -> (id` x) = (id` T))
2	1	dmeqd 3313	. . . . . 6 |- (x = T -> dom (id` x) = dom (id` T))
3		ishoma.1	. . . . . 6 |- O = dom (id` T)
4	2, 3	syl6eqr 1525	. . . . 5 |- (x = T -> dom (id` x) = O)
5	4	eleq2d 1541	. . . 4 |- (x = T -> (a e. dom (id` x) <-> a e. O))
6	4	eleq2d 1541	. . . 4 |- (x = T -> (b e. dom (id` x) <-> b e. O))
7		fveq2 3724	. . . . . . . . . 10 |- (x = T -> (dom` x) = (dom` T))
8	7	dmeqd 3313	. . . . . . . . 9 |- (x = T -> dom (dom` x) = dom (dom` T))
9		ishoma.2	. . . . . . . . 9 |- M = dom (dom` T)
10	8, 9	syl6eqr 1525	. . . . . . . 8 |- (x = T -> dom (dom` x) = M)
11	10	eleq2d 1541	. . . . . . 7 |- (x = T -> (f e. dom (dom` x) <-> f e. M))
12		ishoma.3	. . . . . . . . . 10 |- D = (dom` T)
13	7, 12	syl6eqr 1525	. . . . . . . . 9 |- (x = T -> (dom` x) = D)
14	13	fveq1d 3726	. . . . . . . 8 |- (x = T -> ((dom` x)` f) = (D` f))
15	14	eqeq1d 1483	. . . . . . 7 |- (x = T -> (((dom` x)` f) = a <-> (D` f) = a))
16		fveq2 3724	. . . . . . . . . 10 |- (x = T -> (cod` x) = (cod` T))
17		ishoma.4	. . . . . . . . . 10 |- C = (cod` T)
18	16, 17	syl6eqr 1525	. . . . . . . . 9 |- (x = T -> (cod` x) = C)
19	18	fveq1d 3726	. . . . . . . 8 |- (x = T -> ((cod` x)` f) = (C` f))
20	19	eqeq1d 1483	. . . . . . 7 |- (x = T -> (((cod` x)` f) = b <-> (C` f) = b))
21	11, 15, 20	3anbi123d 893	. . . . . 6 |- (x = T -> ((f e. dom (dom` x) /\ ((dom` x)` f) = a /\ ((cod` x)` f) = b) <-> (f e. M /\ (D` f) = a /\ (C` f) = b)))
22	21	abbidv 1577	. . . . 5 |- (x = T -> {f | (f e. dom (dom` x) /\ ((dom` x)` f) = a /\ ((cod` x)` f) = b)} = {f | (f e. M /\ (D` f) = a /\ (C` f) = b)})
23	22	eqeq2d 1486	. . . 4 |- (x = T -> (c = {f | (f e. dom (dom` x) /\ ((dom` x)` f) = a /\ ((cod` x)` f) = b)} <-> c = {f | (f e. M /\ (D` f) = a /\ (C` f) = b)}))
24	5, 6, 23	3anbi123d 893	. . 3 |- (x = T -> ((a e. dom (id` x) /\ b e. dom (id` x) /\ c = {f | (f e. dom (dom` x) /\ ((dom` x)` f) = a /\ ((cod` x)` f) = b)}) <-> (a e. O /\ b e. O /\ c = {f | (f e. M /\ (D` f) = a /\ (C` f) = b)})))
25	24	oprabbidv 3996	. 2 |- (x = T -> {<.<.a, b>., c>. | (a e. dom (id` x) /\ b e. dom (id` x) /\ c = {f | (f e. dom (dom` x) /\ ((dom` x)` f) = a /\ ((cod` x)` f) = b)})} = {<.<.a, b>., c>. | (a e. O /\ b e. O /\ c = {f | (f e. M /\ (D` f) = a /\ (C` f) = b)})})
26		df-hom 10714	. 2 |- hom = {<.x, y>. | (x e. Cat /\ y = {<.<.a, b>., c>. | (a e. dom (id` x) /\ b e. dom (id` x) /\ c = {f | (f e. dom (dom` x) /\ ((dom` x)` f) = a /\ ((cod` x)` f) = b)})})}
27		fvex 3732	. . . 4 |- (id` T) e. V
28	27	dmex 3360	. . 3 |- dom (id` T) e. V
29	3	eleq2i 1538	. . . . . 6 |- (a e. O <-> a e. dom (id` T))
30	3	eleq2i 1538	. . . . . 6 |- (b e. O <-> b e. dom (id` T))
31		pm4.2 170	. . . . . 6 |- (c = {f | (f e. M /\ (D` f) = a /\ (C` f) = b)} <-> c = {f | (f e. M /\ (D` f) = a /\ (C` f) = b)})
32	29, 30, 31	3anbi123i 822	. . . . 5 |- ((a e. O /\ b e. O /\ c = {f | (f e. M /\ (D` f) = a /\ (C` f) = b)}) <-> (a e. dom (id` T) /\ b e. dom (id` T) /\ c = {f | (f e. M /\ (D` f) = a /\ (C` f) = b)}))
33		df-3an 777	. . . . 5 |- ((a e. dom (id` T) /\ b e. dom (id` T) /\ c = {f | (f e. M /\ (D` f) = a /\ (C` f) = b)}) <-> ((a e. dom (id` T) /\ b e. dom (id` T)) /\ c = {f | (f e. M /\ (D` f) = a /\ (C` f) = b)}))
34	32, 33	bitr 173	. . . 4 |- ((a e. O /\ b e. O /\ c = {f | (f e. M /\ (D` f) = a /\ (C` f) = b)}) <-> ((a e. dom (id` T) /\ b e. dom (id` T)) /\ c = {f | (f e. M /\ (D` f) = a /\ (C` f) = b)}))
35	34	oprabbii 3997	. . 3 |- {<.<.a, b>., c>. | (a e. O /\ b e. O /\ c = {f | (f e. M /\ (D` f) = a /\ (C` f) = b)})} = {<.<.a, b>., c>. | ((a e. dom (id` T) /\ b e. dom (id` T)) /\ c = {f | (f e. M /\ (D` f) = a /\ (C` f) = b)})}
36	28, 28, 35	oprabex2 4021	. 2 |- {<.<.a, b>., c>. | (a e. O /\ b e. O /\ c = {f | (f e. M /\ (D` f) = a /\ (C` f) = b)})} e. V
37	25, 26, 36	fvopab4 3780	1 |- (T e. Cat -> (hom` T) = {<.<.a, b>., c>. | (a e. O /\ b e. O /\ c = {f | (f e. M /\ (D` f) = a /\ (C` f) = b)})})

Syntax hints:   -> wi 3   /\ wa 223   /\ w3a 775   = wceq 956   e. wcel 958  {cab 1463  dom cdm 3170  ` cfv 3182  {copab2 3964  domcdom_ 10644  codccod_ 10645  idcid_ 10646  Catccat 10685  homchom 10713

This theorem is referenced by:  ishomb 10716

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-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-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-fv 3198  df-oprab 3966  df-hom 10714

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