Theorem ishomb 15137

Description: The homset (( hom ` T)` <.A, B>.).

Hypotheses

Ref	Expression
ishomb.1	|- O = dom (id` T)
ishomb.2	|- M = dom (dom` T)
ishomb.3	|- D = (dom` T)
ishomb.4	|- C = (cod` T)
ishomb.5	|- H = ( hom ` T)
ishomb.6	|- T e. Cat

Assertion

Ref	Expression
ishomb	|- ((A e. O /\ B e. O) -> (H` <.A, B>.) = {f | (f e. M /\ (D` f) = A /\ (C` f) = B)})

Distinct variable groups:   A,f   B,f   f,M   T,f

Proof of Theorem ishomb

Step	Hyp	Ref	Expression
1		3anass 862	. . . . 5 |- ((f e. M /\ (D` f) = A /\ (C` f) = B) <-> (f e. M /\ ((D` f) = A /\ (C` f) = B)))
2	1	abbii 2006	. . . 4 |- {f | (f e. M /\ (D` f) = A /\ (C` f) = B)} = {f | (f e. M /\ ((D` f) = A /\ (C` f) = B))}
3		ishomb.2	. . . . . 6 |- M = dom (dom` T)
4		fvex 4689	. . . . . . 7 |- (dom` T) e. _V
5	4	dmex 4208	. . . . . 6 |- dom (dom` T) e. _V
6	3, 5	eqeltri 1967	. . . . 5 |- M e. _V
7	6	zfausab 3459	. . . 4 |- {f | (f e. M /\ ((D` f) = A /\ (C` f) = B))} e. _V
8	2, 7	eqeltri 1967	. . 3 |- {f | (f e. M /\ (D` f) = A /\ (C` f) = B)} e. _V
9		eqeq2 1893	. . . . 5 |- (x = A -> ((D` f) = x <-> (D` f) = A))
10	9	3anbi2d 1173	. . . 4 |- (x = A -> ((f e. M /\ (D` f) = x /\ (C` f) = y) <-> (f e. M /\ (D` f) = A /\ (C` f) = y)))
11	10	abbidv 2008	. . 3 |- (x = A -> {f | (f e. M /\ (D` f) = x /\ (C` f) = y)} = {f | (f e. M /\ (D` f) = A /\ (C` f) = y)})
12		eqeq2 1893	. . . . 5 |- (y = B -> ((C` f) = y <-> (C` f) = B))
13	12	3anbi3d 1174	. . . 4 |- (y = B -> ((f e. M /\ (D` f) = A /\ (C` f) = y) <-> (f e. M /\ (D` f) = A /\ (C` f) = B)))
14	13	abbidv 2008	. . 3 |- (y = B -> {f | (f e. M /\ (D` f) = A /\ (C` f) = y)} = {f | (f e. M /\ (D` f) = A /\ (C` f) = B)})
15		ishomb.5	. . . 4 |- H = ( hom ` T)
16		ishomb.6	. . . . 5 |- T e. Cat
17		ishomb.1	. . . . . . 7 |- O = dom (id` T)
18		ishomb.3	. . . . . . 7 |- D = (dom` T)
19		ishomb.4	. . . . . . 7 |- C = (cod` T)
20	17, 3, 18, 19	ishoma 15136	. . . . . 6 |- (T e. Cat -> ( hom ` T) = {<.<.x, y>., z>. | (x e. O /\ y e. O /\ z = {f | (f e. M /\ (D` f) = x /\ (C` f) = y)})})
21		df-3an 860	. . . . . . . 8 |- ((x e. O /\ y e. O /\ z = {f | (f e. M /\ (D` f) = x /\ (C` f) = y)}) <-> ((x e. O /\ y e. O) /\ z = {f | (f e. M /\ (D` f) = x /\ (C` f) = y)}))
22	21	a1i 8	. . . . . . 7 |- (T e. Cat -> ((x e. O /\ y e. O /\ z = {f | (f e. M /\ (D` f) = x /\ (C` f) = y)}) <-> ((x e. O /\ y e. O) /\ z = {f | (f e. M /\ (D` f) = x /\ (C` f) = y)})))
23	22	oprabbidv 4922	. . . . . 6 |- (T e. Cat -> {<.<.x, y>., z>. | (x e. O /\ y e. O /\ z = {f | (f e. M /\ (D` f) = x /\ (C` f) = y)})} = {<.<.x, y>., z>. | ((x e. O /\ y e. O) /\ z = {f | (f e. M /\ (D` f) = x /\ (C` f) = y)})})
24	20, 23	eqtrd 1925	. . . . 5 |- (T e. Cat -> ( hom ` T) = {<.<.x, y>., z>. | ((x e. O /\ y e. O) /\ z = {f | (f e. M /\ (D` f) = x /\ (C` f) = y)})})
25	16, 24	ax-mp 7	. . . 4 |- ( hom ` T) = {<.<.x, y>., z>. | ((x e. O /\ y e. O) /\ z = {f | (f e. M /\ (D` f) = x /\ (C` f) = y)})}
26	15, 25	eqtri 1908	. . 3 |- H = {<.<.x, y>., z>. | ((x e. O /\ y e. O) /\ z = {f | (f e. M /\ (D` f) = x /\ (C` f) = y)})}
27	8, 11, 14, 26	oprabval2 4957	. 2 |- ((A e. O /\ B e. O) -> (AHB) = {f | (f e. M /\ (D` f) = A /\ (C` f) = B)})
28		df-opr 4886	. 2 |- (AHB) = (H` <.A, B>.)
29	27, 28	syl5eqr 1942	1 |- ((A e. O /\ B e. O) -> (H` <.A, B>.) = {f | (f e. M /\ (D` f) = A /\ (C` f) = B)})

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  {cab 1871  _Vcvv 2292  <.cop 3046  dom cdm 3986  ` cfv 3998  (class class class)co 4884  {copab2 4885  domcdom_ 15059  codccod_ 15060  idcid_ 15061   Cat ccat 15099   hom chom 15134

This theorem is referenced by:  ishomc 15138

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-rex 2110  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-fv 4014  df-opr 4886  df-oprab 4887  df-hom 15135

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