Theorem ishomb 10560

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 777	. . . . 5 |- ((f e. M /\ (D` f) = A /\ (C` f) = B) <-> (f e. M /\ ((D` f) = A /\ (C` f) = B)))
2	1	abbii 1567	. . . 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 3717	. . . . . . 7 |- (dom` T) e. V
5		dmexg 3344	. . . . . . 7 |- ((dom` T) e. V -> dom (dom` T) e. V)
6	4, 5	ax-mp 7	. . . . . 6 |- dom (dom` T) e. V
7	3, 6	eqeltr 1536	. . . . 5 |- M e. V
8	7	zfausab 2713	. . . 4 |- {f | (f e. M /\ ((D` f) = A /\ (C` f) = B))} e. V
9	2, 8	eqeltr 1536	. . 3 |- {f | (f e. M /\ (D` f) = A /\ (C` f) = B)} e. V
10		pm4.2i 171	. . . . 5 |- (x = A -> (f e. M <-> f e. M))
11		id 59	. . . . . 6 |- (x = A -> x = A)
12	11	eqeq2d 1478	. . . . 5 |- (x = A -> ((D` f) = x <-> (D` f) = A))
13		pm4.2i 171	. . . . 5 |- (x = A -> ((C` f) = y <-> (C` f) = y))
14	10, 12, 13	3anbi123d 890	. . . 4 |- (x = A -> ((f e. M /\ (D` f) = x /\ (C` f) = y) <-> (f e. M /\ (D` f) = A /\ (C` f) = y)))
15	14	abbidv 1569	. . 3 |- (x = A -> {f | (f e. M /\ (D` f) = x /\ (C` f) = y)} = {f | (f e. M /\ (D` f) = A /\ (C` f) = y)})
16		pm4.2i 171	. . . . 5 |- (y = B -> (f e. M <-> f e. M))
17		pm4.2i 171	. . . . 5 |- (y = B -> ((D` f) = A <-> (D` f) = A))
18		eqeq2 1476	. . . . 5 |- (y = B -> ((C` f) = y <-> (C` f) = B))
19	16, 17, 18	3anbi123d 890	. . . 4 |- (y = B -> ((f e. M /\ (D` f) = A /\ (C` f) = y) <-> (f e. M /\ (D` f) = A /\ (C` f) = B)))
20	19	abbidv 1569	. . 3 |- (y = B -> {f | (f e. M /\ (D` f) = A /\ (C` f) = y)} = {f | (f e. M /\ (D` f) = A /\ (C` f) = B)})
21		ishomb.5	. . . 4 |- H = (hom` T)
22		ishomb.6	. . . . 5 |- T e. Cat
23		ishomb.1	. . . . . . 7 |- O = dom (id` T)
24		ishomb.3	. . . . . . 7 |- D = (dom` T)
25		ishomb.4	. . . . . . 7 |- C = (cod` T)
26	23, 3, 24, 25	ishoma 10559	. . . . . 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)})})
27		df-3an 775	. . . . . . . 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)}))
28	27	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)})))
29	28	oprabbidv 3981	. . . . . 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)})})
30	26, 29	eqtrd 1499	. . . . 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)})})
31	22, 30	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)})}
32	21, 31	eqtr 1487	. . 3 |- H = {<.<.x, y>., z>. | ((x e. O /\ y e. O) /\ z = {f | (f e. M /\ (D` f) = x /\ (C` f) = y)})}
33	9, 15, 20, 32	oprabval2 4013	. 2 |- ((A e. O /\ B e. O) -> (AHB) = {f | (f e. M /\ (D` f) = A /\ (C` f) = B)})
34		df-opr 3950	. 2 |- (AHB) = (H` <.A, B>.)
35	33, 34	syl5eqr 1513	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 146   /\ wa 223   /\ w3a 773   = wceq 953   e. wcel 955  {cab 1456  Vcvv 1802  <.cop 2401  dom cdm 3160  ` cfv 3172  (class class class)co 3948  {copab2 3949  domcdom_ 10488  codccod_ 10489  idcid_ 10490  Catccat 10529  homchom 10557

This theorem is referenced by:  ishomc 10561

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-sbc 1932  df-csb 1992  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-opr 3950  df-oprab 3951  df-hom 10558

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