Theorem cotr 3432

Description: Two ways of saying a relation is transitive. Definition of transitivity in [Schechter] p. 51.

Assertion

Ref	Expression
cotr	|- ((R o. R) (_ R <-> A.xA.yA.z((xRy /\ yRz) -> xRz))

Distinct variable group:   x,y,z,R

Proof of Theorem cotr

Step	Hyp	Ref	Expression
1		ssel 2060	. . . . . . . 8 |- ({<.x, z>. | E.y(xRy /\ yRz)} (_ R -> (<.x, z>. e. {<.x, z>. | E.y(xRy /\ yRz)} -> <.x, z>. e. R))
2		df-br 2616	. . . . . . . 8 |- (xRz <-> <.x, z>. e. R)
3	1, 2	syl6ibr 213	. . . . . . 7 |- ({<.x, z>. | E.y(xRy /\ yRz)} (_ R -> (<.x, z>. e. {<.x, z>. | E.y(xRy /\ yRz)} -> xRz))
4		opabid 2806	. . . . . . 7 |- (<.x, z>. e. {<.x, z>. | E.y(xRy /\ yRz)} <-> E.y(xRy /\ yRz))
5	3, 4	syl5ibr 207	. . . . . 6 |- ({<.x, z>. | E.y(xRy /\ yRz)} (_ R -> (E.y(xRy /\ yRz) -> xRz))
6		df-co 3183	. . . . . . 7 |- (R o. R) = {<.x, z>. | E.y(xRy /\ yRz)}
7	6	sseq1i 2082	. . . . . 6 |- ((R o. R) (_ R <-> {<.x, z>. | E.y(xRy /\ yRz)} (_ R)
8		19.23v 1292	. . . . . 6 |- (A.y((xRy /\ yRz) -> xRz) <-> (E.y(xRy /\ yRz) -> xRz))
9	5, 7, 8	3imtr4 219	. . . . 5 |- ((R o. R) (_ R -> A.y((xRy /\ yRz) -> xRz))
10	9	19.21aiv 1285	. . . 4 |- ((R o. R) (_ R -> A.zA.y((xRy /\ yRz) -> xRz))
11		alcom 1031	. . . 4 |- (A.yA.z((xRy /\ yRz) -> xRz) <-> A.zA.y((xRy /\ yRz) -> xRz))
12	10, 11	sylibr 200	. . 3 |- ((R o. R) (_ R -> A.yA.z((xRy /\ yRz) -> xRz))
13	12	19.21aiv 1285	. 2 |- ((R o. R) (_ R -> A.xA.yA.z((xRy /\ yRz) -> xRz))
14		ssopab2 2818	. . . . 5 |- ({<.x, z>. | E.y(xRy /\ yRz)} (_ {<.x, z>. | xRz} <-> A.xA.z(E.y(xRy /\ yRz) -> xRz))
15	8	albii 998	. . . . . . 7 |- (A.zA.y((xRy /\ yRz) -> xRz) <-> A.z(E.y(xRy /\ yRz) -> xRz))
16	11, 15	bitr 173	. . . . . 6 |- (A.yA.z((xRy /\ yRz) -> xRz) <-> A.z(E.y(xRy /\ yRz) -> xRz))
17	16	albii 998	. . . . 5 |- (A.xA.yA.z((xRy /\ yRz) -> xRz) <-> A.xA.z(E.y(xRy /\ yRz) -> xRz))
18	14, 17	bitr4 176	. . . 4 |- ({<.x, z>. | E.y(xRy /\ yRz)} (_ {<.x, z>. | xRz} <-> A.xA.yA.z((xRy /\ yRz) -> xRz))
19		opabss 2664	. . . . 5 |- {<.x, z>. | xRz} (_ R
20		sstr2 2068	. . . . 5 |- ({<.x, z>. | E.y(xRy /\ yRz)} (_ {<.x, z>. | xRz} -> ({<.x, z>. | xRz} (_ R -> {<.x, z>. | E.y(xRy /\ yRz)} (_ R))
21	19, 20	mpi 44	. . . 4 |- ({<.x, z>. | E.y(xRy /\ yRz)} (_ {<.x, z>. | xRz} -> {<.x, z>. | E.y(xRy /\ yRz)} (_ R)
22	18, 21	sylbir 201	. . 3 |- (A.xA.yA.z((xRy /\ yRz) -> xRz) -> {<.x, z>. | E.y(xRy /\ yRz)} (_ R)
23	22, 6	syl5ss 2102	. 2 |- (A.xA.yA.z((xRy /\ yRz) -> xRz) -> (R o. R) (_ R)
24	13, 23	impbi 157	1 |- ((R o. R) (_ R <-> A.xA.yA.z((xRy /\ yRz) -> xRz))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223  A.wal 953   e. wcel 957  E.wex 979   (_ wss 2044  <.cop 2408   class class class wbr 2615  {copab 2662   o. ccom 3170

This theorem is referenced by:  dfer2 4255  pslem 8605

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-9 964  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458  ax-sep 2699  ax-pow 2738  ax-pr 2775

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-v 1809  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-nul 2278  df-pw 2399  df-sn 2409  df-pr 2410  df-op 2413  df-br 2616  df-opab 2663  df-co 3183

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