Theorem caovord 7571

Description: Convert an operation ordering law to class notation. (Contributed by NM, 19-Feb-1996.)

Hypotheses

Ref	Expression
caovord.1	⊢ 𝐴 ∈ V
caovord.2	⊢ 𝐵 ∈ V
caovord.3	⊢ (𝑧 ∈ 𝑆 → (𝑥𝑅𝑦 ↔ (𝑧𝐹𝑥)𝑅(𝑧𝐹𝑦)))

Assertion

Ref	Expression
caovord	⊢ (𝐶 ∈ 𝑆 → (𝐴𝑅𝐵 ↔ (𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵)))

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

Proof of Theorem caovord

Step	Hyp	Ref	Expression
1		oveq1 7367	. . . 4 ⊢ (𝑧 = 𝐶 → (𝑧𝐹𝐴) = (𝐶𝐹𝐴))
2		oveq1 7367	. . . 4 ⊢ (𝑧 = 𝐶 → (𝑧𝐹𝐵) = (𝐶𝐹𝐵))
3	1, 2	breq12d 5088	. . 3 ⊢ (𝑧 = 𝐶 → ((𝑧𝐹𝐴)𝑅(𝑧𝐹𝐵) ↔ (𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵)))
4	3	bibi2d 344	. 2 ⊢ (𝑧 = 𝐶 → ((𝐴𝑅𝐵 ↔ (𝑧𝐹𝐴)𝑅(𝑧𝐹𝐵)) ↔ (𝐴𝑅𝐵 ↔ (𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵))))
5		caovord.1	. . 3 ⊢ 𝐴 ∈ V
6		caovord.2	. . 3 ⊢ 𝐵 ∈ V
7		breq1 5078	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥𝑅𝑦 ↔ 𝐴𝑅𝑦))
8		oveq2 7368	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑧𝐹𝑥) = (𝑧𝐹𝐴))
9	8	breq1d 5085	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((𝑧𝐹𝑥)𝑅(𝑧𝐹𝑦) ↔ (𝑧𝐹𝐴)𝑅(𝑧𝐹𝑦)))
10	7, 9	bibi12d 347	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑥𝑅𝑦 ↔ (𝑧𝐹𝑥)𝑅(𝑧𝐹𝑦)) ↔ (𝐴𝑅𝑦 ↔ (𝑧𝐹𝐴)𝑅(𝑧𝐹𝑦))))
11		breq2 5079	. . . . . 6 ⊢ (𝑦 = 𝐵 → (𝐴𝑅𝑦 ↔ 𝐴𝑅𝐵))
12		oveq2 7368	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (𝑧𝐹𝑦) = (𝑧𝐹𝐵))
13	12	breq2d 5087	. . . . . 6 ⊢ (𝑦 = 𝐵 → ((𝑧𝐹𝐴)𝑅(𝑧𝐹𝑦) ↔ (𝑧𝐹𝐴)𝑅(𝑧𝐹𝐵)))
14	11, 13	bibi12d 347	. . . . 5 ⊢ (𝑦 = 𝐵 → ((𝐴𝑅𝑦 ↔ (𝑧𝐹𝐴)𝑅(𝑧𝐹𝑦)) ↔ (𝐴𝑅𝐵 ↔ (𝑧𝐹𝐴)𝑅(𝑧𝐹𝐵))))
15	10, 14	sylan9bb 515	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ((𝑥𝑅𝑦 ↔ (𝑧𝐹𝑥)𝑅(𝑧𝐹𝑦)) ↔ (𝐴𝑅𝐵 ↔ (𝑧𝐹𝐴)𝑅(𝑧𝐹𝐵))))
16	15	imbi2d 342	. . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ((𝑧 ∈ 𝑆 → (𝑥𝑅𝑦 ↔ (𝑧𝐹𝑥)𝑅(𝑧𝐹𝑦))) ↔ (𝑧 ∈ 𝑆 → (𝐴𝑅𝐵 ↔ (𝑧𝐹𝐴)𝑅(𝑧𝐹𝐵)))))
17		caovord.3	. . 3 ⊢ (𝑧 ∈ 𝑆 → (𝑥𝑅𝑦 ↔ (𝑧𝐹𝑥)𝑅(𝑧𝐹𝑦)))
18	5, 6, 16, 17	vtocl2 3512	. 2 ⊢ (𝑧 ∈ 𝑆 → (𝐴𝑅𝐵 ↔ (𝑧𝐹𝐴)𝑅(𝑧𝐹𝐵)))
19	4, 18	vtoclga 3522	1 ⊢ (𝐶 ∈ 𝑆 → (𝐴𝑅𝐵 ↔ (𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 397   = wceq 1548   ∈ wcel 2121  Vcvv 3433   class class class wbr 5075  (class class class)co 7360

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-rab 3394  df-v 3435  df-dif 3888  df-un 3890  df-ss 3902  df-nul 4265  df-if 4458  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-br 5076  df-iota 6445  df-fv 6497  df-ov 7363

This theorem is referenced by:  caovord2  7572  caovord3  7573  genpcl  10926