Theorem vtoclr 5608

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	. . . . 5 ⊢ Rel 𝑅
2	1	brrelex12i 5600	. . . 4 ⊢ (𝐴𝑅𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
3	1	brrelex2i 5602	. . . 4 ⊢ (𝐵𝑅𝐶 → 𝐶 ∈ V)
4		breq1 5062	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝑥𝑅𝑦 ↔ 𝐴𝑅𝑦))
5	4	anbi1d 631	. . . . . . 7 ⊢ (𝑥 = 𝐴 → ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝐶) ↔ (𝐴𝑅𝑦 ∧ 𝑦𝑅𝐶)))
6		breq1 5062	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑥𝑅𝐶 ↔ 𝐴𝑅𝐶))
7	5, 6	imbi12d 347	. . . . . 6 ⊢ (𝑥 = 𝐴 → (((𝑥𝑅𝑦 ∧ 𝑦𝑅𝐶) → 𝑥𝑅𝐶) ↔ ((𝐴𝑅𝑦 ∧ 𝑦𝑅𝐶) → 𝐴𝑅𝐶)))
8	7	imbi2d 343	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝐶 ∈ V → ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝐶) → 𝑥𝑅𝐶)) ↔ (𝐶 ∈ V → ((𝐴𝑅𝑦 ∧ 𝑦𝑅𝐶) → 𝐴𝑅𝐶))))
9		breq2 5063	. . . . . . . 8 ⊢ (𝑦 = 𝐵 → (𝐴𝑅𝑦 ↔ 𝐴𝑅𝐵))
10		breq1 5062	. . . . . . . 8 ⊢ (𝑦 = 𝐵 → (𝑦𝑅𝐶 ↔ 𝐵𝑅𝐶))
11	9, 10	anbi12d 632	. . . . . . 7 ⊢ (𝑦 = 𝐵 → ((𝐴𝑅𝑦 ∧ 𝑦𝑅𝐶) ↔ (𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶)))
12	11	imbi1d 344	. . . . . 6 ⊢ (𝑦 = 𝐵 → (((𝐴𝑅𝑦 ∧ 𝑦𝑅𝐶) → 𝐴𝑅𝐶) ↔ ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶)))
13	12	imbi2d 343	. . . . 5 ⊢ (𝑦 = 𝐵 → ((𝐶 ∈ V → ((𝐴𝑅𝑦 ∧ 𝑦𝑅𝐶) → 𝐴𝑅𝐶)) ↔ (𝐶 ∈ V → ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶))))
14		breq2 5063	. . . . . . . 8 ⊢ (𝑧 = 𝐶 → (𝑦𝑅𝑧 ↔ 𝑦𝑅𝐶))
15	14	anbi2d 630	. . . . . . 7 ⊢ (𝑧 = 𝐶 → ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) ↔ (𝑥𝑅𝑦 ∧ 𝑦𝑅𝐶)))
16		breq2 5063	. . . . . . 7 ⊢ (𝑧 = 𝐶 → (𝑥𝑅𝑧 ↔ 𝑥𝑅𝐶))
17	15, 16	imbi12d 347	. . . . . 6 ⊢ (𝑧 = 𝐶 → (((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧) ↔ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝐶) → 𝑥𝑅𝐶)))
18		vtoclr.2	. . . . . 6 ⊢ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)
19	17, 18	vtoclg 3564	. . . . 5 ⊢ (𝐶 ∈ V → ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝐶) → 𝑥𝑅𝐶))
20	8, 13, 19	vtocl2g 3569	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐶 ∈ V → ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶)))
21	2, 3, 20	syl2im 40	. . 3 ⊢ (𝐴𝑅𝐵 → (𝐵𝑅𝐶 → ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶)))
22	21	imp 409	. 2 ⊢ ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶))
23	22	pm2.43i 52	1 ⊢ ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶)

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1536   ∈ wcel 2113  Vcvv 3491   class class class wbr 5059  Rel wrel 5553

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2792  ax-sep 5196  ax-nul 5203  ax-pr 5323

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-clab 2799  df-cleq 2813  df-clel 2892  df-nfc 2962  df-ral 3142  df-rex 3143  df-rab 3146  df-v 3493  df-dif 3932  df-un 3934  df-in 3936  df-ss 3945  df-nul 4285  df-if 4461  df-sn 4561  df-pr 4563  df-op 4567  df-br 5060  df-opab 5122  df-xp 5554  df-rel 5555

This theorem is referenced by:  domtr  8555