Theorem ishomb 10598

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 778	. . . . 5 |- ((f e. M /\ (D` f) = A /\ (C` f) = B) <-> (f e. M /\ ((D` f) = A /\ (C` f) = B)))
2	1	abbii 1572	. . . 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 3724	. . . . . . 7 |- (dom` T) e. V
5	4	dmex 3354	. . . . . 6 |- dom (dom` T) e. V
6	3, 5	eqeltr 1541	. . . . 5 |- M e. V
7	6	zfausab 2718	. . . 4 |- {f | (f e. M /\ ((D` f) = A /\ (C` f) = B))} e. V
8	2, 7	eqeltr 1541	. . 3 |- {f | (f e. M /\ (D` f) = A /\ (C` f) = B)} e. V
9		pm4.2i 171	. . . . 5 |- (x = A -> (f e. M <-> f e. M))
10		id 59	. . . . . 6 |- (x = A -> x = A)
11	10	eqeq2d 1483	. . . . 5 |- (x = A -> ((D` f) = x <-> (D` f) = A))
12		pm4.2i 171	. . . . 5 |- (x = A -> ((C` f) = y <-> (C` f) = y))
13	9, 11, 12	3anbi123d 891	. . . 4 |- (x = A -> ((f e. M /\ (D` f) = x /\ (C` f) = y) <-> (f e. M /\ (D` f) = A /\ (C` f) = y)))
14	13	abbidv 1574	. . 3 |- (x = A -> {f | (f e. M /\ (D` f) = x /\ (C` f) = y)} = {f | (f e. M /\ (D` f) = A /\ (C` f) = y)})
15		pm4.2i 171	. . . . 5 |- (y = B -> (f e. M <-> f e. M))
16		pm4.2i 171	. . . . 5 |- (y = B -> ((D` f) = A <-> (D` f) = A))
17		eqeq2 1481	. . . . 5 |- (y = B -> ((C` f) = y <-> (C` f) = B))
18	15, 16, 17	3anbi123d 891	. . . 4 |- (y = B -> ((f e. M /\ (D` f) = A /\ (C` f) = y) <-> (f e. M /\ (D` f) = A /\ (C` f) = B)))
19	18	abbidv 1574	. . 3 |- (y = B -> {f | (f e. M /\ (D` f) = A /\ (C` f) = y)} = {f | (f e. M /\ (D` f) = A /\ (C` f) = B)})
20		ishomb.5	. . . 4 |- H = (hom` T)
21		ishomb.6	. . . . 5 |- T e. Cat
22		ishomb.1	. . . . . . 7 |- O = dom (id` T)
23		ishomb.3	. . . . . . 7 |- D = (dom` T)
24		ishomb.4	. . . . . . 7 |- C = (cod` T)
25	22, 3, 23, 24	ishoma 10597	. . . . . 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)})})
26		df-3an 776	. . . . . . . 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)}))
27	26	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)})))
28	27	oprabbidv 3988	. . . . . 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)})})
29	25, 28	eqtrd 1504	. . . . 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)})})
30	21, 29	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)})}
31	20, 30	eqtr 1492	. . 3 |- H = {<.<.x, y>., z>. | ((x e. O /\ y e. O) /\ z = {f | (f e. M /\ (D` f) = x /\ (C` f) = y)})}
32	8, 14, 19, 31	oprabval2 4020	. 2 |- ((A e. O /\ B e. O) -> (AHB) = {f | (f e. M /\ (D` f) = A /\ (C` f) = B)})
33		df-opr 3957	. 2 |- (AHB) = (H` <.A, B>.)
34	32, 33	syl5eqr 1518	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 774   = wceq 954   e. wcel 956  {cab 1461  Vcvv 1807  <.cop 2407  dom cdm 3165  ` cfv 3177  (class class class)co 3955  {copab2 3956  domcdom_ 10526  codccod_ 10527  idcid_ 10528  Catccat 10567  homchom 10595

This theorem is referenced by:  ishomc 10599

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-sbc 1938  df-csb 1998  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-opr 3957  df-oprab 3958  df-hom 10596

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