Theorem ishoma 10559

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 3709	. . . . . . 7 |- (x = T -> (id` x) = (id` T))
2	1	dmeqd 3302	. . . . . 6 |- (x = T -> dom (id` x) = dom (id` T))
3		ishoma.1	. . . . . 6 |- O = dom (id` T)
4	2, 3	syl6eqr 1517	. . . . 5 |- (x = T -> dom (id` x) = O)
5	4	eleq2d 1533	. . . 4 |- (x = T -> (a e. dom (id` x) <-> a e. O))
6	4	eleq2d 1533	. . . 4 |- (x = T -> (b e. dom (id` x) <-> b e. O))
7		fveq2 3709	. . . . . . . . . 10 |- (x = T -> (dom` x) = (dom` T))
8	7	dmeqd 3302	. . . . . . . . 9 |- (x = T -> dom (dom` x) = dom (dom` T))
9		ishoma.2	. . . . . . . . 9 |- M = dom (dom` T)
10	8, 9	syl6eqr 1517	. . . . . . . 8 |- (x = T -> dom (dom` x) = M)
11	10	eleq2d 1533	. . . . . . 7 |- (x = T -> (f e. dom (dom` x) <-> f e. M))
12		ishoma.3	. . . . . . . . . 10 |- D = (dom` T)
13	7, 12	syl6eqr 1517	. . . . . . . . 9 |- (x = T -> (dom` x) = D)
14	13	fveq1d 3711	. . . . . . . 8 |- (x = T -> ((dom` x)` f) = (D` f))
15	14	eqeq1d 1475	. . . . . . 7 |- (x = T -> (((dom` x)` f) = a <-> (D` f) = a))
16		fveq2 3709	. . . . . . . . . 10 |- (x = T -> (cod` x) = (cod` T))
17		ishoma.4	. . . . . . . . . 10 |- C = (cod` T)
18	16, 17	syl6eqr 1517	. . . . . . . . 9 |- (x = T -> (cod` x) = C)
19	18	fveq1d 3711	. . . . . . . 8 |- (x = T -> ((cod` x)` f) = (C` f))
20	19	eqeq1d 1475	. . . . . . 7 |- (x = T -> (((cod` x)` f) = b <-> (C` f) = b))
21	11, 15, 20	3anbi123d 890	. . . . . 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 1569	. . . . 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 1478	. . . 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 890	. . 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 3981	. 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 10558	. 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 3717	. . . 4 |- (id` T) e. V
28		dmexg 3344	. . . 4 |- ((id` T) e. V -> dom (id` T) e. V)
29	27, 28	ax-mp 7	. . 3 |- dom (id` T) e. V
30	3	eleq2i 1530	. . . . . 6 |- (a e. O <-> a e. dom (id` T))
31	3	eleq2i 1530	. . . . . 6 |- (b e. O <-> b e. dom (id` T))
32		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)})
33	30, 31, 32	3anbi123i 820	. . . . 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)}))
34		df-3an 775	. . . . 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)}))
35	33, 34	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)}))
36	35	oprabbii 3982	. . 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)})}
37	29, 29, 36	oprabex2 4006	. 2 |- {<.<.a, b>., c>. | (a e. O /\ b e. O /\ c = {f | (f e. M /\ (D` f) = a /\ (C` f) = b)})} e. V
38	25, 26, 37	fvopab4 3765	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 773   = wceq 953   e. wcel 955  {cab 1456  Vcvv 1802  dom cdm 3160  ` cfv 3172  {copab2 3949  domcdom_ 10488  codccod_ 10489  idcid_ 10490  Catccat 10529  homchom 10557

This theorem is referenced by:  ishomb 10560

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-rep 2683  ax-sep 2693  ax-pow 2732  ax-pr 2769  ax-un 2857

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-rex 1642  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-uni 2494  df-br 2610  df-opab 2657  df-id 2824  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-fv 3188  df-oprab 3951  df-hom 10558

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