Theorem cotr 3420

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 2053	. . . . . . . 8 |- ({<.x, z>. | E.y(xRy /\ yRz)} (_ R -> (<.x, z>. e. {<.x, z>. | E.y(xRy /\ yRz)} -> <.x, z>. e. R))
2		df-br 2610	. . . . . . . 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 2799	. . . . . . 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 3177	. . . . . . 7 |- (R o. R) = {<.x, z>. | E.y(xRy /\ yRz)}
7	6	sseq1i 2075	. . . . . 6 |- ((R o. R) (_ R <-> {<.x, z>. | E.y(xRy /\ yRz)} (_ R)
8		19.23v 1288	. . . . . 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 1281	. . . 4 |- ((R o. R) (_ R -> A.zA.y((xRy /\ yRz) -> xRz))
11		alcom 1028	. . . 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 1281	. 2 |- ((R o. R) (_ R -> A.xA.yA.z((xRy /\ yRz) -> xRz))
14		ssopab2 2811	. . . . 5 |- ({<.x, z>. | E.y(xRy /\ yRz)} (_ {<.x, z>. | xRz} <-> A.xA.z(E.y(xRy /\ yRz) -> xRz))
15	8	albii 996	. . . . . . 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 996	. . . . 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 2658	. . . . 5 |- {<.x, z>. | xRz} (_ R
20		sstr2 2061	. . . . 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 2095	. 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 951   e. wcel 955  E.wex 977   (_ wss 2037  <.cop 2401   class class class wbr 2609  {copab 2656   o. ccom 3164

This theorem is referenced by:  dfer2 4246  pslem 8573

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-pow 2732  ax-pr 2769

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-br 2610  df-opab 2657  df-co 3177

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