Theorem cotr 2625

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(s):   x,y,z,R

Proof of Theorem cotr

Step	Hyp	Ref	Expression
1		ssel 1502	. . . . . . . 8 |- ({<.x, z>. | E.y(xRy /\ yRz)} (_ R -> (<.x, z>. e. {<.x, z>. | E.y(xRy /\ yRz)} -> <.x, z>. e. R))
2		df-br 2063	. . . . . . . 8 |- (xRz <-> <.x, z>. e. R)
3	1, 2	syl6ibr 186	. . . . . . 7 |- ({<.x, z>. | E.y(xRy /\ yRz)} (_ R -> (<.x, z>. e. {<.x, z>. | E.y(xRy /\ yRz)} -> xRz))
4		opabid 2099	. . . . . . 7 |- (<.x, z>. e. {<.x, z>. | E.y(xRy /\ yRz)} <-> E.y(xRy /\ yRz))
5	3, 4	syl5ibr 182	. . . . . 6 |- ({<.x, z>. | E.y(xRy /\ yRz)} (_ R -> (E.y(xRy /\ yRz) -> xRz))
6		df-co 2427	. . . . . . 7 |- (R o. R) = {<.x, z>. | E.y(xRy /\ yRz)}
7	6	sseq1i 1524	. . . . . 6 |- ((R o. R) (_ R <-> {<.x, z>. | E.y(xRy /\ yRz)} (_ R)
8		19.23v 950	. . . . . 6 |- (A.y((xRy /\ yRz) -> xRz) <-> (E.y(xRy /\ yRz) -> xRz))
9	5, 7, 8	3imtr4 192	. . . . 5 |- ((R o. R) (_ R -> A.y((xRy /\ yRz) -> xRz))
10	9	19.21aiv 943	. . . 4 |- ((R o. R) (_ R -> A.zA.y((xRy /\ yRz) -> xRz))
11		alcom 715	. . . 4 |- (A.yA.z((xRy /\ yRz) -> xRz) <-> A.zA.y((xRy /\ yRz) -> xRz))
12	10, 11	sylibr 175	. . 3 |- ((R o. R) (_ R -> A.yA.z((xRy /\ yRz) -> xRz))
13	12	19.21aiv 943	. 2 |- ((R o. R) (_ R -> A.xA.yA.z((xRy /\ yRz) -> xRz))
14		ssopab2 2119	. . . . 5 |- ({<.x, z>. | E.y(xRy /\ yRz)} (_ {<.x, z>. | xRz} <-> A.xA.z(E.y(xRy /\ yRz) -> xRz))
15	8	bial 695	. . . . . . 7 |- (A.zA.y((xRy /\ yRz) -> xRz) <-> A.z(E.y(xRy /\ yRz) -> xRz))
16	11, 15	bitr 151	. . . . . 6 |- (A.yA.z((xRy /\ yRz) -> xRz) <-> A.z(E.y(xRy /\ yRz) -> xRz))
17	16	bial 695	. . . . 5 |- (A.xA.yA.z((xRy /\ yRz) -> xRz) <-> A.xA.z(E.y(xRy /\ yRz) -> xRz))
18	14, 17	bitr4 154	. . . 4 |- ({<.x, z>. | E.y(xRy /\ yRz)} (_ {<.x, z>. | xRz} <-> A.xA.yA.z((xRy /\ yRz) -> xRz))
19		opabss 2100	. . . . 5 |- {<.x, z>. | xRz} (_ R
20		sstr2 1510	. . . . 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 176	. . 3 |- (A.xA.yA.z((xRy /\ yRz) -> xRz) -> {<.x, z>. | E.y(xRy /\ yRz)} (_ R)
23	22, 6	syl5ss 1544	. 2 |- (A.xA.yA.z((xRy /\ yRz) -> xRz) -> (R o. R) (_ R)
24	13, 23	impbi 139	1 |- ((R o. R) (_ R <-> A.xA.yA.z((xRy /\ yRz) -> xRz))

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196  A.wal 672  E.wex 678   e. wcel 1092   (_ wss 1487  <.cop 1810   class class class wbr 2054  {copab 2055   o. ccom 2414

This theorem is referenced by:  er2 3201

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-opab 2098  df-co 2427

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