Theorem trrd 5925

Description: Deduce transitivity from its properties. (Contributed by SF, 22-Feb-2015.)

Hypotheses

Ref	Expression
trrd.1	|- (ph -> R e. V)
trrd.2	|- (ph -> A e. W)
trrd.3	|- ((ph /\ (x e. A /\ y e. A /\ z e. A) /\ (xRy /\ yRz)) -> xRz)

Assertion

Ref	Expression
trrd	|- (ph -> R Trans A)

Distinct variable groups:   x,A,y,z   ph,x,y,z   x,R,y,z

Allowed substitution hints:   V(x,y,z)   W(x,y,z)

Proof of Theorem trrd

Dummy variables a r are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-3an 936	. . . . . 6 |- ((x e. A /\ y e. A /\ z e. A) <-> ((x e. A /\ y e. A) /\ z e. A))
2		trrd.3	. . . . . . 7 |- ((ph /\ (x e. A /\ y e. A /\ z e. A) /\ (xRy /\ yRz)) -> xRz)
3	2	3exp 1150	. . . . . 6 |- (ph -> ((x e. A /\ y e. A /\ z e. A) -> ((xRy /\ yRz) -> xRz)))
4	1, 3	syl5bir 209	. . . . 5 |- (ph -> (((x e. A /\ y e. A) /\ z e. A) -> ((xRy /\ yRz) -> xRz)))
5	4	exp3a 425	. . . 4 |- (ph -> ((x e. A /\ y e. A) -> (z e. A -> ((xRy /\ yRz) -> xRz))))
6	5	ralrimdv 2703	. . 3 |- (ph -> ((x e. A /\ y e. A) -> A.z e. A ((xRy /\ yRz) -> xRz)))
7	6	ralrimivv 2705	. 2 |- (ph -> A.x e. A A.y e. A A.z e. A ((xRy /\ yRz) -> xRz))
8		trrd.1	. . 3 |- (ph -> R e. V)
9		trrd.2	. . 3 |- (ph -> A e. W)
10		breq 4641	. . . . . . . 8 |- (r = R -> (xry <-> xRy))
11		breq 4641	. . . . . . . 8 |- (r = R -> (yrz <-> yRz))
12	10, 11	anbi12d 691	. . . . . . 7 |- (r = R -> ((xry /\ yrz) <-> (xRy /\ yRz)))
13		breq 4641	. . . . . . 7 |- (r = R -> (xrz <-> xRz))
14	12, 13	imbi12d 311	. . . . . 6 |- (r = R -> (((xry /\ yrz) -> xrz) <-> ((xRy /\ yRz) -> xRz)))
15	14	ralbidv 2634	. . . . 5 |- (r = R -> (A.z e. a ((xry /\ yrz) -> xrz) <-> A.z e. a ((xRy /\ yRz) -> xRz)))
16	15	2ralbidv 2656	. . . 4 |- (r = R -> (A.x e. a A.y e. a A.z e. a ((xry /\ yrz) -> xrz) <-> A.x e. a A.y e. a A.z e. a ((xRy /\ yRz) -> xRz)))
17		raleq 2807	. . . . . 6 |- (a = A -> (A.z e. a ((xRy /\ yRz) -> xRz) <-> A.z e. A ((xRy /\ yRz) -> xRz)))
18	17	raleqbi1dv 2815	. . . . 5 |- (a = A -> (A.y e. a A.z e. a ((xRy /\ yRz) -> xRz) <-> A.y e. A A.z e. A ((xRy /\ yRz) -> xRz)))
19	18	raleqbi1dv 2815	. . . 4 |- (a = A -> (A.x e. a A.y e. a A.z e. a ((xRy /\ yRz) -> xRz) <-> A.x e. A A.y e. A A.z e. A ((xRy /\ yRz) -> xRz)))
20		df-trans 5899	. . . 4 |- Trans = {<.r, a>. | A.x e. a A.y e. a A.z e. a ((xry /\ yrz) -> xrz)}
21	16, 19, 20	brabg 4706	. . 3 |- ((R e. V /\ A e. W) -> (R Trans A <-> A.x e. A A.y e. A A.z e. A ((xRy /\ yRz) -> xRz)))
22	8, 9, 21	syl2anc 642	. 2 |- (ph -> (R Trans A <-> A.x e. A A.y e. A A.z e. A ((xRy /\ yRz) -> xRz)))
23	7, 22	mpbird 223	1 |- (ph -> R Trans A)

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358   /\ w3a 934   = wceq 1642   e. wcel 1710  A.wral 2614   class class class wbr 4639   Trans ctrans 5888

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-trans 5899

This theorem is referenced by:  pod  5936  iserd  5942