Theorem ishoma 10597

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 3716	. . . . . . 7 |- (x = T -> (id` x) = (id` T))
2	1	dmeqd 3308	. . . . . 6 |- (x = T -> dom (id` x) = dom (id` T))
3		ishoma.1	. . . . . 6 |- O = dom (id` T)
4	2, 3	syl6eqr 1522	. . . . 5 |- (x = T -> dom (id` x) = O)
5	4	eleq2d 1538	. . . 4 |- (x = T -> (a e. dom (id` x) <-> a e. O))
6	4	eleq2d 1538	. . . 4 |- (x = T -> (b e. dom (id` x) <-> b e. O))
7		fveq2 3716	. . . . . . . . . 10 |- (x = T -> (dom` x) = (dom` T))
8	7	dmeqd 3308	. . . . . . . . 9 |- (x = T -> dom (dom` x) = dom (dom` T))
9		ishoma.2	. . . . . . . . 9 |- M = dom (dom` T)
10	8, 9	syl6eqr 1522	. . . . . . . 8 |- (x = T -> dom (dom` x) = M)
11	10	eleq2d 1538	. . . . . . 7 |- (x = T -> (f e. dom (dom` x) <-> f e. M))
12		ishoma.3	. . . . . . . . . 10 |- D = (dom` T)
13	7, 12	syl6eqr 1522	. . . . . . . . 9 |- (x = T -> (dom` x) = D)
14	13	fveq1d 3718	. . . . . . . 8 |- (x = T -> ((dom` x)` f) = (D` f))
15	14	eqeq1d 1480	. . . . . . 7 |- (x = T -> (((dom` x)` f) = a <-> (D` f) = a))
16		fveq2 3716	. . . . . . . . . 10 |- (x = T -> (cod` x) = (cod` T))
17		ishoma.4	. . . . . . . . . 10 |- C = (cod` T)
18	16, 17	syl6eqr 1522	. . . . . . . . 9 |- (x = T -> (cod` x) = C)
19	18	fveq1d 3718	. . . . . . . 8 |- (x = T -> ((cod` x)` f) = (C` f))
20	19	eqeq1d 1480	. . . . . . 7 |- (x = T -> (((cod` x)` f) = b <-> (C` f) = b))
21	11, 15, 20	3anbi123d 891	. . . . . 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 1574	. . . . 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 1483	. . . 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 891	. . 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 3988	. 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 10596	. 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 3724	. . . 4 |- (id` T) e. V
28	27	dmex 3354	. . 3 |- dom (id` T) e. V
29	3	eleq2i 1535	. . . . . 6 |- (a e. O <-> a e. dom (id` T))
30	3	eleq2i 1535	. . . . . 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 821	. . . . 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 776	. . . . 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 3989	. . 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 4013	. 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 3772	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 774   = wceq 954   e. wcel 956  {cab 1461  dom cdm 3165  ` cfv 3177  {copab2 3956  domcdom_ 10526  codccod_ 10527  idcid_ 10528  Catccat 10567  homchom 10595

This theorem is referenced by:  ishomb 10598

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-rep 2688  ax-sep 2698  ax-pow 2737  ax-pr 2774  ax-un 2861

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-rex 1647  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-uni 2499  df-br 2615  df-opab 2662  df-id 2830  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fun 3187  df-fn 3188  df-fv 3193  df-oprab 3958  df-hom 10596

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