Theorem cotr 2625

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

Assertion

Ref	Expression
cotr	⊢ ((R ∘ R) ⊆ R ↔ ∀x∀y∀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⟩∣∃y(xRy ∧ yRz)} ⊆ R → (⟨x, z⟩ ∈ {⟨x, z⟩∣∃y(xRy ∧ yRz)} → ⟨x, z⟩ ∈ R))
2		df-br 2063	. . . . . . . 8 ⊢ (xRz ↔ ⟨x, z⟩ ∈ R)
3	1, 2	syl6ibr 186	. . . . . . 7 ⊢ ({⟨x, z⟩∣∃y(xRy ∧ yRz)} ⊆ R → (⟨x, z⟩ ∈ {⟨x, z⟩∣∃y(xRy ∧ yRz)} → xRz))
4		opabid 2099	. . . . . . 7 ⊢ (⟨x, z⟩ ∈ {⟨x, z⟩∣∃y(xRy ∧ yRz)} ↔ ∃y(xRy ∧ yRz))
5	3, 4	syl5ibr 182	. . . . . 6 ⊢ ({⟨x, z⟩∣∃y(xRy ∧ yRz)} ⊆ R → (∃y(xRy ∧ yRz) → xRz))
6		df-co 2427	. . . . . . 7 ⊢ (R ∘ R) = {⟨x, z⟩∣∃y(xRy ∧ yRz)}
7	6	sseq1i 1524	. . . . . 6 ⊢ ((R ∘ R) ⊆ R ↔ {⟨x, z⟩∣∃y(xRy ∧ yRz)} ⊆ R)
8		19.23v 950	. . . . . 6 ⊢ (∀y((xRy ∧ yRz) → xRz) ↔ (∃y(xRy ∧ yRz) → xRz))
9	5, 7, 8	3imtr4 192	. . . . 5 ⊢ ((R ∘ R) ⊆ R → ∀y((xRy ∧ yRz) → xRz))
10	9	19.21aiv 943	. . . 4 ⊢ ((R ∘ R) ⊆ R → ∀z∀y((xRy ∧ yRz) → xRz))
11		alcom 715	. . . 4 ⊢ (∀y∀z((xRy ∧ yRz) → xRz) ↔ ∀z∀y((xRy ∧ yRz) → xRz))
12	10, 11	sylibr 175	. . 3 ⊢ ((R ∘ R) ⊆ R → ∀y∀z((xRy ∧ yRz) → xRz))
13	12	19.21aiv 943	. 2 ⊢ ((R ∘ R) ⊆ R → ∀x∀y∀z((xRy ∧ yRz) → xRz))
14		ssopab2 2119	. . . . 5 ⊢ ({⟨x, z⟩∣∃y(xRy ∧ yRz)} ⊆ {⟨x, z⟩∣xRz} ↔ ∀x∀z(∃y(xRy ∧ yRz) → xRz))
15	8	bial 695	. . . . . . 7 ⊢ (∀z∀y((xRy ∧ yRz) → xRz) ↔ ∀z(∃y(xRy ∧ yRz) → xRz))
16	11, 15	bitr 151	. . . . . 6 ⊢ (∀y∀z((xRy ∧ yRz) → xRz) ↔ ∀z(∃y(xRy ∧ yRz) → xRz))
17	16	bial 695	. . . . 5 ⊢ (∀x∀y∀z((xRy ∧ yRz) → xRz) ↔ ∀x∀z(∃y(xRy ∧ yRz) → xRz))
18	14, 17	bitr4 154	. . . 4 ⊢ ({⟨x, z⟩∣∃y(xRy ∧ yRz)} ⊆ {⟨x, z⟩∣xRz} ↔ ∀x∀y∀z((xRy ∧ yRz) → xRz))
19		opabss 2100	. . . . 5 ⊢ {⟨x, z⟩∣xRz} ⊆ R
20		sstr2 1510	. . . . 5 ⊢ ({⟨x, z⟩∣∃y(xRy ∧ yRz)} ⊆ {⟨x, z⟩∣xRz} → ({⟨x, z⟩∣xRz} ⊆ R → {⟨x, z⟩∣∃y(xRy ∧ yRz)} ⊆ R))
21	19, 20	mpi 44	. . . 4 ⊢ ({⟨x, z⟩∣∃y(xRy ∧ yRz)} ⊆ {⟨x, z⟩∣xRz} → {⟨x, z⟩∣∃y(xRy ∧ yRz)} ⊆ R)
22	18, 21	sylbir 176	. . 3 ⊢ (∀x∀y∀z((xRy ∧ yRz) → xRz) → {⟨x, z⟩∣∃y(xRy ∧ yRz)} ⊆ R)
23	22, 6	syl5ss 1544	. 2 ⊢ (∀x∀y∀z((xRy ∧ yRz) → xRz) → (R ∘ R) ⊆ R)
24	13, 23	impbi 139	1 ⊢ ((R ∘ R) ⊆ R ↔ ∀x∀y∀z((xRy ∧ yRz) → xRz))

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196  ∀wal 672  ∃wex 678   ∈ wcel 1092   ⊆ wss 1487  ⟨cop 1810   class class class wbr 2054  {copab 2055   ∘ 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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