Theorem ishomb 10687

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 781	. . . . 5 |- ((f e. M /\ (D` f) = A /\ (C` f) = B) <-> (f e. M /\ ((D` f) = A /\ (C` f) = B)))
2	1	abbii 1578	. . . 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 3738	. . . . . . 7 |- (dom` T) e. V
5	4	dmex 3366	. . . . . 6 |- dom (dom` T) e. V
6	3, 5	eqeltr 1547	. . . . 5 |- M e. V
7	6	zfausab 2728	. . . 4 |- {f | (f e. M /\ ((D` f) = A /\ (C` f) = B))} e. V
8	2, 7	eqeltr 1547	. . 3 |- {f | (f e. M /\ (D` f) = A /\ (C` f) = B)} e. V
9		eqeq2 1487	. . . . 5 |- (x = A -> ((D` f) = x <-> (D` f) = A))
10	9	3anbi2d 900	. . . 4 |- (x = A -> ((f e. M /\ (D` f) = x /\ (C` f) = y) <-> (f e. M /\ (D` f) = A /\ (C` f) = y)))
11	10	abbidv 1580	. . 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 1487	. . . . 5 |- (y = B -> ((C` f) = y <-> (C` f) = B))
13	12	3anbi3d 901	. . . 4 |- (y = B -> ((f e. M /\ (D` f) = A /\ (C` f) = y) <-> (f e. M /\ (D` f) = A /\ (C` f) = B)))
14	13	abbidv 1580	. . 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 10686	. . . . . 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 779	. . . . . . . 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 4002	. . . . . 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 1510	. . . . 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	eqtr 1498	. . 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 4034	. 2 |- ((A e. O /\ B e. O) -> (AHB) = {f | (f e. M /\ (D` f) = A /\ (C` f) = B)})
28		df-opr 3971	. 2 |- (AHB) = (H` <.A, B>.)
29	27, 28	syl5eqr 1524	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 777   = wceq 958   e. wcel 960  {cab 1466  Vcvv 1814  <.cop 2415  dom cdm 3176  ` cfv 3188  (class class class)co 3969  {copab2 3970  domcdom_ 10615  codccod_ 10616  idcid_ 10617  Catccat 10656  homchom 10684

This theorem is referenced by:  ishomc 10688

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2698  ax-sep 2708  ax-pow 2748  ax-pr 2785  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-rex 1653  df-v 1815  df-sbc 1945  df-csb 2005  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-fv 3204  df-opr 3971  df-oprab 3972  df-hom 10685

Copyright terms: Public domain