Theorem ishomc 10688

Description: The predicate F e. ((hom` T)` <.A, B>.) JFM vol. 1.2 p. 411 th. 18.

Hypotheses

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

Assertion

Ref	Expression
ishomc	|- ((A e. O /\ B e. O) -> (F e. (H` <.A, B>.) <-> (F e. M /\ (D` F) = A /\ (C` F) = B)))

Proof of Theorem ishomc

Step	Hyp	Ref	Expression
1		ishomc.1	. . . 4 |- O = dom (id` T)
2		ishomc.2	. . . 4 |- M = dom (dom` T)
3		ishomc.3	. . . 4 |- D = (dom` T)
4		ishomc.4	. . . 4 |- C = (cod` T)
5		ishomc.5	. . . 4 |- H = (hom` T)
6		ishomc.6	. . . 4 |- T e. Cat
7	1, 2, 3, 4, 5, 6	ishomb 10687	. . 3 |- ((A e. O /\ B e. O) -> (H` <.A, B>.) = {x | (x e. M /\ (D` x) = A /\ (C` x) = B)})
8	7	eleq2d 1544	. 2 |- ((A e. O /\ B e. O) -> (F e. (H` <.A, B>.) <-> F e. {x | (x e. M /\ (D` x) = A /\ (C` x) = B)}))
9		3simp1 790	. . 3 |- ((F e. M /\ (D` F) = A /\ (C` F) = B) -> F e. M)
10		eleq1 1537	. . . . 5 |- (x = F -> (x e. M <-> F e. M))
11		fveq2 3730	. . . . . 6 |- (x = F -> (D` x) = (D` F))
12	11	eqeq1d 1486	. . . . 5 |- (x = F -> ((D` x) = A <-> (D` F) = A))
13		fveq2 3730	. . . . . 6 |- (x = F -> (C` x) = (C` F))
14	13	eqeq1d 1486	. . . . 5 |- (x = F -> ((C` x) = B <-> (C` F) = B))
15	10, 12, 14	3anbi123d 895	. . . 4 |- (x = F -> ((x e. M /\ (D` x) = A /\ (C` x) = B) <-> (F e. M /\ (D` F) = A /\ (C` F) = B)))
16	15	elab3g 1905	. . 3 |- (((F e. M /\ (D` F) = A /\ (C` F) = B) -> F e. M) -> (F e. {x | (x e. M /\ (D` x) = A /\ (C` x) = B)} <-> (F e. M /\ (D` F) = A /\ (C` F) = B)))
17	9, 16	ax-mp 7	. 2 |- (F e. {x | (x e. M /\ (D` x) = A /\ (C` x) = B)} <-> (F e. M /\ (D` F) = A /\ (C` F) = B))
18	8, 17	syl6bb 538	1 |- ((A e. O /\ B e. O) -> (F e. (H` <.A, B>.) <-> (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  <.cop 2415  dom cdm 3176  ` cfv 3188  domcdom_ 10615  codccod_ 10616  idcid_ 10617  Catccat 10656  homchom 10684

This theorem is referenced by:  ishomd 10689

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

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