Theorem vtoclr 5164

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	brrelexi 5158	. . . . 5 ⊢ (𝐴𝑅𝐵 → 𝐴 ∈ V)
3	1	brrelex2i 5159	. . . . 5 ⊢ (𝐴𝑅𝐵 → 𝐵 ∈ V)
4	2, 3	jca 554	. . . 4 ⊢ (𝐴𝑅𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
5	1	brrelex2i 5159	. . . 4 ⊢ (𝐵𝑅𝐶 → 𝐶 ∈ V)
6		breq1 4656	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝑥𝑅𝑦 ↔ 𝐴𝑅𝑦))
7	6	anbi1d 741	. . . . . . 7 ⊢ (𝑥 = 𝐴 → ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝐶) ↔ (𝐴𝑅𝑦 ∧ 𝑦𝑅𝐶)))
8		breq1 4656	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑥𝑅𝐶 ↔ 𝐴𝑅𝐶))
9	7, 8	imbi12d 334	. . . . . 6 ⊢ (𝑥 = 𝐴 → (((𝑥𝑅𝑦 ∧ 𝑦𝑅𝐶) → 𝑥𝑅𝐶) ↔ ((𝐴𝑅𝑦 ∧ 𝑦𝑅𝐶) → 𝐴𝑅𝐶)))
10	9	imbi2d 330	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝐶 ∈ V → ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝐶) → 𝑥𝑅𝐶)) ↔ (𝐶 ∈ V → ((𝐴𝑅𝑦 ∧ 𝑦𝑅𝐶) → 𝐴𝑅𝐶))))
11		breq2 4657	. . . . . . . 8 ⊢ (𝑦 = 𝐵 → (𝐴𝑅𝑦 ↔ 𝐴𝑅𝐵))
12		breq1 4656	. . . . . . . 8 ⊢ (𝑦 = 𝐵 → (𝑦𝑅𝐶 ↔ 𝐵𝑅𝐶))
13	11, 12	anbi12d 747	. . . . . . 7 ⊢ (𝑦 = 𝐵 → ((𝐴𝑅𝑦 ∧ 𝑦𝑅𝐶) ↔ (𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶)))
14	13	imbi1d 331	. . . . . 6 ⊢ (𝑦 = 𝐵 → (((𝐴𝑅𝑦 ∧ 𝑦𝑅𝐶) → 𝐴𝑅𝐶) ↔ ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶)))
15	14	imbi2d 330	. . . . 5 ⊢ (𝑦 = 𝐵 → ((𝐶 ∈ V → ((𝐴𝑅𝑦 ∧ 𝑦𝑅𝐶) → 𝐴𝑅𝐶)) ↔ (𝐶 ∈ V → ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶))))
16		breq2 4657	. . . . . . . 8 ⊢ (𝑧 = 𝐶 → (𝑦𝑅𝑧 ↔ 𝑦𝑅𝐶))
17	16	anbi2d 740	. . . . . . 7 ⊢ (𝑧 = 𝐶 → ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) ↔ (𝑥𝑅𝑦 ∧ 𝑦𝑅𝐶)))
18		breq2 4657	. . . . . . 7 ⊢ (𝑧 = 𝐶 → (𝑥𝑅𝑧 ↔ 𝑥𝑅𝐶))
19	17, 18	imbi12d 334	. . . . . 6 ⊢ (𝑧 = 𝐶 → (((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧) ↔ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝐶) → 𝑥𝑅𝐶)))
20		vtoclr.2	. . . . . 6 ⊢ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)
21	19, 20	vtoclg 3266	. . . . 5 ⊢ (𝐶 ∈ V → ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝐶) → 𝑥𝑅𝐶))
22	10, 15, 21	vtocl2g 3270	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐶 ∈ V → ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶)))
23	4, 5, 22	syl2im 40	. . 3 ⊢ (𝐴𝑅𝐵 → (𝐵𝑅𝐶 → ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶)))
24	23	imp 445	. 2 ⊢ ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶))
25	24	pm2.43i 52	1 ⊢ ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶)

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990  Vcvv 3200   class class class wbr 4653  Rel wrel 5119

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121

This theorem is referenced by:  domtr  8009