Theorem vtoclr 4797

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 4792	. . . . 5 ⊢ (𝐴𝑅𝐵 → 𝐴 ∈ V)
3	1	brrelex2i 4793	. . . . 5 ⊢ (𝐴𝑅𝐵 → 𝐵 ∈ V)
4	2, 3	jca 306	. . . 4 ⊢ (𝐴𝑅𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
5	1	brrelex2i 4793	. . . 4 ⊢ (𝐵𝑅𝐶 → 𝐶 ∈ V)
6		breq1 4111	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝑥𝑅𝑦 ↔ 𝐴𝑅𝑦))
7	6	anbi1d 465	. . . . . . 7 ⊢ (𝑥 = 𝐴 → ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝐶) ↔ (𝐴𝑅𝑦 ∧ 𝑦𝑅𝐶)))
8		breq1 4111	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑥𝑅𝐶 ↔ 𝐴𝑅𝐶))
9	7, 8	imbi12d 234	. . . . . 6 ⊢ (𝑥 = 𝐴 → (((𝑥𝑅𝑦 ∧ 𝑦𝑅𝐶) → 𝑥𝑅𝐶) ↔ ((𝐴𝑅𝑦 ∧ 𝑦𝑅𝐶) → 𝐴𝑅𝐶)))
10	9	imbi2d 230	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝐶 ∈ V → ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝐶) → 𝑥𝑅𝐶)) ↔ (𝐶 ∈ V → ((𝐴𝑅𝑦 ∧ 𝑦𝑅𝐶) → 𝐴𝑅𝐶))))
11		breq2 4112	. . . . . . . 8 ⊢ (𝑦 = 𝐵 → (𝐴𝑅𝑦 ↔ 𝐴𝑅𝐵))
12		breq1 4111	. . . . . . . 8 ⊢ (𝑦 = 𝐵 → (𝑦𝑅𝐶 ↔ 𝐵𝑅𝐶))
13	11, 12	anbi12d 473	. . . . . . 7 ⊢ (𝑦 = 𝐵 → ((𝐴𝑅𝑦 ∧ 𝑦𝑅𝐶) ↔ (𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶)))
14	13	imbi1d 231	. . . . . 6 ⊢ (𝑦 = 𝐵 → (((𝐴𝑅𝑦 ∧ 𝑦𝑅𝐶) → 𝐴𝑅𝐶) ↔ ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶)))
15	14	imbi2d 230	. . . . 5 ⊢ (𝑦 = 𝐵 → ((𝐶 ∈ V → ((𝐴𝑅𝑦 ∧ 𝑦𝑅𝐶) → 𝐴𝑅𝐶)) ↔ (𝐶 ∈ V → ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶))))
16		breq2 4112	. . . . . . . 8 ⊢ (𝑧 = 𝐶 → (𝑦𝑅𝑧 ↔ 𝑦𝑅𝐶))
17	16	anbi2d 464	. . . . . . 7 ⊢ (𝑧 = 𝐶 → ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) ↔ (𝑥𝑅𝑦 ∧ 𝑦𝑅𝐶)))
18		breq2 4112	. . . . . . 7 ⊢ (𝑧 = 𝐶 → (𝑥𝑅𝑧 ↔ 𝑥𝑅𝐶))
19	17, 18	imbi12d 234	. . . . . 6 ⊢ (𝑧 = 𝐶 → (((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧) ↔ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝐶) → 𝑥𝑅𝐶)))
20		vtoclr.2	. . . . . 6 ⊢ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)
21	19, 20	vtoclg 2874	. . . . 5 ⊢ (𝐶 ∈ V → ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝐶) → 𝑥𝑅𝐶))
22	10, 15, 21	vtocl2g 2878	. . . 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 124	. 2 ⊢ ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶))
25	24	pm2.43i 49	1 ⊢ ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1398   ∈ wcel 2203  Vcvv 2812   class class class wbr 4108  Rel wrel 4753

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4227  ax-pow 4286  ax-pr 4321

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2814  df-un 3214  df-in 3216  df-ss 3223  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-br 4109  df-opab 4171  df-xp 4754  df-rel 4755

This theorem is referenced by:  domtr  7024