Theorem catded 10697

Description: A category is a deductive system.

Assertion

Ref	Expression
catded	|- (T e. Cat -> T e. Ded)

Proof of Theorem catded

Step	Hyp	Ref	Expression
1		relcat 10694	. . . . 5 |- Rel Cat
2		reldcat 10695	. . . . 5 |- Rel dom Cat
3		relrcat 10696	. . . . 5 |- Rel ran Cat
4	1, 2, 3	3pm3.2i 818	. . . 4 |- (Rel Cat /\ Rel dom Cat /\ Rel ran Cat)
5		11st22nd 10458	. . . 4 |- (((Rel Cat /\ Rel dom Cat /\ Rel ran Cat) /\ T e. Cat) -> T = <.<.(1st` (1st` T)), (2nd` (1st` T))>., <.(1st` (2nd` T)), (2nd` (2nd` T))>.>.)
6	4, 5	mpan 695	. . 3 |- (T e. Cat -> T = <.<.(1st` (1st` T)), (2nd` (1st` T))>., <.(1st` (2nd` T)), (2nd` (2nd` T))>.>.)
7		eqid 1475	. . . . . 6 |- (dom` T) = (dom` T)
8	7	domval 10655	. . . . 5 |- (dom` T) = (1st` (1st` T))
9		eqid 1475	. . . . . 6 |- (cod` T) = (cod` T)
10	9	codval 10656	. . . . 5 |- (cod` T) = (2nd` (1st` T))
11	8, 10	opeq12i 2492	. . . 4 |- <.(dom` T), (cod` T)>. = <.(1st` (1st` T)), (2nd` (1st` T))>.
12		eqid 1475	. . . . . 6 |- (id` T) = (id` T)
13	12	idval 10657	. . . . 5 |- (id` T) = (1st` (2nd` T))
14		eqid 1475	. . . . . 6 |- (o` T) = (o` T)
15	14	cmpval 10658	. . . . 5 |- (o` T) = (2nd` (2nd` T))
16	13, 15	opeq12i 2492	. . . 4 |- <.(id` T), (o` T)>. = <.(1st` (2nd` T)), (2nd` (2nd` T))>.
17	11, 16	opeq12i 2492	. . 3 |- <.<.(dom` T), (cod` T)>., <.(id` T), (o` T)>.>. = <.<.(1st` (1st` T)), (2nd` (1st` T))>., <.(1st` (2nd` T)), (2nd` (2nd` T))>.>.
18	6, 17	syl6eqr 1525	. 2 |- (T e. Cat -> T = <.<.(dom` T), (cod` T)>., <.(id` T), (o` T)>.>.)
19		eqid 1475	. . . . 5 |- dom (dom` T) = dom (dom` T)
20		eqid 1475	. . . . 5 |- dom (id` T) = dom (id` T)
21	7, 9, 12, 14, 19, 20	cati 10688	. . . 4 |- (T e. Cat -> ((<.<.(dom` T), (cod` T)>., <.(id` T), (o` T)>.>. e. Ded /\ A.x e. dom (dom` T)A.y e. dom (dom` T)A.z e. dom (dom` T)((((dom` T)` z) = ((cod` T)` y) /\ ((dom` T)` y) = ((cod` T)` x)) -> (z(o` T)(y(o` T)x)) = ((z(o` T)y)(o` T)x))) /\ (A.w e. dom (id` T)A.x e. dom (dom` T)(((cod` T)` x) = w -> (((id` T)` w)(o` T)x) = x) /\ A.w e. dom (id` T)A.x e. dom (dom` T)(((dom` T)` x) = w -> (x(o` T)((id` T)` w)) = x))))
22	21	pm3.26d 321	. . 3 |- (T e. Cat -> (<.<.(dom` T), (cod` T)>., <.(id` T), (o` T)>.>. e. Ded /\ A.x e. dom (dom` T)A.y e. dom (dom` T)A.z e. dom (dom` T)((((dom` T)` z) = ((cod` T)` y) /\ ((dom` T)` y) = ((cod` T)` x)) -> (z(o` T)(y(o` T)x)) = ((z(o` T)y)(o` T)x))))
23	22	pm3.26d 321	. 2 |- (T e. Cat -> <.<.(dom` T), (cod` T)>., <.(id` T), (o` T)>.>. e. Ded)
24	18, 23	eqeltrd 1548	1 |- (T e. Cat -> T e. Ded)

Syntax hints:   -> wi 3   /\ wa 223   /\ w3a 775   = wceq 956   e. wcel 958  A.wral 1645  <.cop 2411  dom cdm 3170  ran crn 3171  Rel wrel 3175  ` cfv 3182  (class class class)co 3963  1stc1st 4077  2ndc2nd 4078  domcdom_ 10644  codccod_ 10645  idcid_ 10646  oco_ 10647  Dedcded 10667  Catccat 10685

This theorem is referenced by:  domc 10698  codc 10699  idc 10700  cmppfc 10701  idosc 10702  cmppfcd 10703  domcmpc 10704  codcmpc 10705  mrdmcd 10722  eqidob 10723  homib 10724  homgrf 10730

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-int 2534  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-fo 3196  df-fv 3198  df-opr 3965  df-1st 4079  df-2nd 4080  df-doma 10649  df-coda 10650  df-ida 10651  df-cmpa 10652  df-cat 10686

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