Theorem cotr 5027

Description: Two ways of saying a relation is transitive. Definition of transitivity in [Schechter] p. 51. (The proof was shortened by Andrew Salmon, 27-Aug-2011.) (Contributed by set.mm contributors, 27-Dec-1996.) (Revised by set.mm contributors, 27-Aug-2011.)

Assertion

Ref	Expression
cotr	⊢ ((R ∘ R) ⊆ R ↔ ∀x∀y∀z((xRy ∧ yRz) → xRz))

Distinct variable group:   x,y,z,R

Proof of Theorem cotr

Step	Hyp	Ref	Expression
1		ssrel 4845	. 2 ⊢ ((R ∘ R) ⊆ R ↔ ∀x∀z(⟨x, z⟩ ∈ (R ∘ R) → ⟨x, z⟩ ∈ R))
2		alcom 1737	. . . 4 ⊢ (∀y∀z((xRy ∧ yRz) → xRz) ↔ ∀z∀y((xRy ∧ yRz) → xRz))
3		19.23v 1891	. . . . . 6 ⊢ (∀y((xRy ∧ yRz) → xRz) ↔ (∃y(xRy ∧ yRz) → xRz))
4		brco 4884	. . . . . . . 8 ⊢ (x(R ∘ R)z ↔ ∃y(xRy ∧ yRz))
5		df-br 4641	. . . . . . . 8 ⊢ (x(R ∘ R)z ↔ ⟨x, z⟩ ∈ (R ∘ R))
6	4, 5	bitr3i 242	. . . . . . 7 ⊢ (∃y(xRy ∧ yRz) ↔ ⟨x, z⟩ ∈ (R ∘ R))
7		df-br 4641	. . . . . . 7 ⊢ (xRz ↔ ⟨x, z⟩ ∈ R)
8	6, 7	imbi12i 316	. . . . . 6 ⊢ ((∃y(xRy ∧ yRz) → xRz) ↔ (⟨x, z⟩ ∈ (R ∘ R) → ⟨x, z⟩ ∈ R))
9	3, 8	bitri 240	. . . . 5 ⊢ (∀y((xRy ∧ yRz) → xRz) ↔ (⟨x, z⟩ ∈ (R ∘ R) → ⟨x, z⟩ ∈ R))
10	9	albii 1566	. . . 4 ⊢ (∀z∀y((xRy ∧ yRz) → xRz) ↔ ∀z(⟨x, z⟩ ∈ (R ∘ R) → ⟨x, z⟩ ∈ R))
11	2, 10	bitri 240	. . 3 ⊢ (∀y∀z((xRy ∧ yRz) → xRz) ↔ ∀z(⟨x, z⟩ ∈ (R ∘ R) → ⟨x, z⟩ ∈ R))
12	11	albii 1566	. 2 ⊢ (∀x∀y∀z((xRy ∧ yRz) → xRz) ↔ ∀x∀z(⟨x, z⟩ ∈ (R ∘ R) → ⟨x, z⟩ ∈ R))
13	1, 12	bitr4i 243	1 ⊢ ((R ∘ R) ⊆ R ↔ ∀x∀y∀z((xRy ∧ yRz) → xRz))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540  ∃wex 1541   ∈ wcel 1710   ⊆ wss 3258  ⟨cop 4562   class class class wbr 4640   ∘ ccom 4722

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

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 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-co 4727

This theorem is referenced by: (None)