Theorem vtoclr 4659

Description: Variable to class conversion of transitive relation. (Contributed by NM, 9-Jun-1998.) (Revised by Mario Carneiro, 26-Apr-2015.)

Hypotheses

Ref	Expression
vtoclr.1	⊢ Rel 𝑅
vtoclr.2	⊢ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)

Assertion

Ref	Expression
vtoclr	⊢ ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶)

Distinct variable groups:   𝑥,𝑦,𝐴   𝑦,𝐵   𝑥,𝑧,𝐶,𝑦   𝑥,𝑅,𝑦,𝑧

Allowed substitution hints:   𝐴(𝑧)   𝐵(𝑥,𝑧)

Proof of Theorem vtoclr

Step	Hyp	Ref	Expression
1		vtoclr.1	. . . . . 6 ⊢ Rel 𝑅
2	1	brrelex1i 4654	. . . . 5 ⊢ (𝐴𝑅𝐵 → 𝐴 ∈ V)
3	1	brrelex2i 4655	. . . . 5 ⊢ (𝐴𝑅𝐵 → 𝐵 ∈ V)
4	2, 3	jca 304	. . . 4 ⊢ (𝐴𝑅𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
5	1	brrelex2i 4655	. . . 4 ⊢ (𝐵𝑅𝐶 → 𝐶 ∈ V)
6		breq1 3992	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝑥𝑅𝑦 ↔ 𝐴𝑅𝑦))
7	6	anbi1d 462	. . . . . . 7 ⊢ (𝑥 = 𝐴 → ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝐶) ↔ (𝐴𝑅𝑦 ∧ 𝑦𝑅𝐶)))
8		breq1 3992	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑥𝑅𝐶 ↔ 𝐴𝑅𝐶))
9	7, 8	imbi12d 233	. . . . . 6 ⊢ (𝑥 = 𝐴 → (((𝑥𝑅𝑦 ∧ 𝑦𝑅𝐶) → 𝑥𝑅𝐶) ↔ ((𝐴𝑅𝑦 ∧ 𝑦𝑅𝐶) → 𝐴𝑅𝐶)))
10	9	imbi2d 229	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝐶 ∈ V → ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝐶) → 𝑥𝑅𝐶)) ↔ (𝐶 ∈ V → ((𝐴𝑅𝑦 ∧ 𝑦𝑅𝐶) → 𝐴𝑅𝐶))))
11		breq2 3993	. . . . . . . 8 ⊢ (𝑦 = 𝐵 → (𝐴𝑅𝑦 ↔ 𝐴𝑅𝐵))
12		breq1 3992	. . . . . . . 8 ⊢ (𝑦 = 𝐵 → (𝑦𝑅𝐶 ↔ 𝐵𝑅𝐶))
13	11, 12	anbi12d 470	. . . . . . 7 ⊢ (𝑦 = 𝐵 → ((𝐴𝑅𝑦 ∧ 𝑦𝑅𝐶) ↔ (𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶)))
14	13	imbi1d 230	. . . . . 6 ⊢ (𝑦 = 𝐵 → (((𝐴𝑅𝑦 ∧ 𝑦𝑅𝐶) → 𝐴𝑅𝐶) ↔ ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶)))
15	14	imbi2d 229	. . . . 5 ⊢ (𝑦 = 𝐵 → ((𝐶 ∈ V → ((𝐴𝑅𝑦 ∧ 𝑦𝑅𝐶) → 𝐴𝑅𝐶)) ↔ (𝐶 ∈ V → ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶))))
16		breq2 3993	. . . . . . . 8 ⊢ (𝑧 = 𝐶 → (𝑦𝑅𝑧 ↔ 𝑦𝑅𝐶))
17	16	anbi2d 461	. . . . . . 7 ⊢ (𝑧 = 𝐶 → ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) ↔ (𝑥𝑅𝑦 ∧ 𝑦𝑅𝐶)))
18		breq2 3993	. . . . . . 7 ⊢ (𝑧 = 𝐶 → (𝑥𝑅𝑧 ↔ 𝑥𝑅𝐶))
19	17, 18	imbi12d 233	. . . . . 6 ⊢ (𝑧 = 𝐶 → (((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧) ↔ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝐶) → 𝑥𝑅𝐶)))
20		vtoclr.2	. . . . . 6 ⊢ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)
21	19, 20	vtoclg 2790	. . . . 5 ⊢ (𝐶 ∈ V → ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝐶) → 𝑥𝑅𝐶))
22	10, 15, 21	vtocl2g 2794	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐶 ∈ V → ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶)))
23	4, 5, 22	syl2im 38	. . 3 ⊢ (𝐴𝑅𝐵 → (𝐵𝑅𝐶 → ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶)))
24	23	imp 123	. 2 ⊢ ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶))
25	24	pm2.43i 49	1 ⊢ ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶)

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1348   ∈ wcel 2141  Vcvv 2730   class class class wbr 3989  Rel wrel 4616

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-xp 4617  df-rel 4618

This theorem is referenced by:  domtr  6763