Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  ovi3 GIF version

Theorem ovi3 5637
 Description: The value of an operation class abstraction. Special case. (Contributed by NM, 28-May-1995.) (Revised by Mario Carneiro, 29-Dec-2014.)
Hypotheses
Ref Expression
ovi3.1 (((𝐴𝐻𝐵𝐻) ∧ (𝐶𝐻𝐷𝐻)) → 𝑆 ∈ (𝐻 × 𝐻))
ovi3.2 (((𝑤 = 𝐴𝑣 = 𝐵) ∧ (𝑢 = 𝐶𝑓 = 𝐷)) → 𝑅 = 𝑆)
ovi3.3 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ (𝐻 × 𝐻) ∧ 𝑦 ∈ (𝐻 × 𝐻)) ∧ ∃𝑤𝑣𝑢𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅))}
Assertion
Ref Expression
ovi3 (((𝐴𝐻𝐵𝐻) ∧ (𝐶𝐻𝐷𝐻)) → (⟨𝐴, 𝐵𝐹𝐶, 𝐷⟩) = 𝑆)
Distinct variable groups:   𝑢,𝑓,𝑣,𝑤,𝑥,𝑦,𝑧,𝐴   𝐵,𝑓,𝑢,𝑣,𝑤,𝑥,𝑦,𝑧   𝑥,𝑅,𝑦,𝑧   𝐶,𝑓,𝑢,𝑣,𝑤,𝑦,𝑧   𝐷,𝑓,𝑢,𝑣,𝑤,𝑦,𝑧   𝑓,𝐻,𝑢,𝑣,𝑤,𝑥,𝑦,𝑧   𝑆,𝑓,𝑢,𝑣,𝑤,𝑧
Allowed substitution hints:   𝐶(𝑥)   𝐷(𝑥)   𝑅(𝑤,𝑣,𝑢,𝑓)   𝑆(𝑥,𝑦)   𝐹(𝑥,𝑦,𝑧,𝑤,𝑣,𝑢,𝑓)

Proof of Theorem ovi3
StepHypRef Expression
1 ovi3.1 . . . 4 (((𝐴𝐻𝐵𝐻) ∧ (𝐶𝐻𝐷𝐻)) → 𝑆 ∈ (𝐻 × 𝐻))
2 elex 2566 . . . 4 (𝑆 ∈ (𝐻 × 𝐻) → 𝑆 ∈ V)
31, 2syl 14 . . 3 (((𝐴𝐻𝐵𝐻) ∧ (𝐶𝐻𝐷𝐻)) → 𝑆 ∈ V)
4 isset 2561 . . 3 (𝑆 ∈ V ↔ ∃𝑧 𝑧 = 𝑆)
53, 4sylib 127 . 2 (((𝐴𝐻𝐵𝐻) ∧ (𝐶𝐻𝐷𝐻)) → ∃𝑧 𝑧 = 𝑆)
6 nfv 1421 . . 3 𝑧((𝐴𝐻𝐵𝐻) ∧ (𝐶𝐻𝐷𝐻))
7 nfcv 2178 . . . . 5 𝑧𝐴, 𝐵
8 ovi3.3 . . . . . 6 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ (𝐻 × 𝐻) ∧ 𝑦 ∈ (𝐻 × 𝐻)) ∧ ∃𝑤𝑣𝑢𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅))}
9 nfoprab3 5556 . . . . . 6 𝑧{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ (𝐻 × 𝐻) ∧ 𝑦 ∈ (𝐻 × 𝐻)) ∧ ∃𝑤𝑣𝑢𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅))}
108, 9nfcxfr 2175 . . . . 5 𝑧𝐹
11 nfcv 2178 . . . . 5 𝑧𝐶, 𝐷
127, 10, 11nfov 5535 . . . 4 𝑧(⟨𝐴, 𝐵𝐹𝐶, 𝐷⟩)
1312nfeq1 2187 . . 3 𝑧(⟨𝐴, 𝐵𝐹𝐶, 𝐷⟩) = 𝑆
14 ovi3.2 . . . . . . 7 (((𝑤 = 𝐴𝑣 = 𝐵) ∧ (𝑢 = 𝐶𝑓 = 𝐷)) → 𝑅 = 𝑆)
1514eqeq2d 2051 . . . . . 6 (((𝑤 = 𝐴𝑣 = 𝐵) ∧ (𝑢 = 𝐶𝑓 = 𝐷)) → (𝑧 = 𝑅𝑧 = 𝑆))
1615copsex4g 3984 . . . . 5 (((𝐴𝐻𝐵𝐻) ∧ (𝐶𝐻𝐷𝐻)) → (∃𝑤𝑣𝑢𝑓((⟨𝐴, 𝐵⟩ = ⟨𝑤, 𝑣⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅) ↔ 𝑧 = 𝑆))
17 opelxpi 4376 . . . . . 6 ((𝐴𝐻𝐵𝐻) → ⟨𝐴, 𝐵⟩ ∈ (𝐻 × 𝐻))
18 opelxpi 4376 . . . . . 6 ((𝐶𝐻𝐷𝐻) → ⟨𝐶, 𝐷⟩ ∈ (𝐻 × 𝐻))
19 nfcv 2178 . . . . . . 7 𝑥𝐴, 𝐵
20 nfcv 2178 . . . . . . 7 𝑦𝐴, 𝐵
21 nfcv 2178 . . . . . . 7 𝑦𝐶, 𝐷
22 nfv 1421 . . . . . . . 8 𝑥𝑤𝑣𝑢𝑓((⟨𝐴, 𝐵⟩ = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅)
23 nfoprab1 5554 . . . . . . . . . . 11 𝑥{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ (𝐻 × 𝐻) ∧ 𝑦 ∈ (𝐻 × 𝐻)) ∧ ∃𝑤𝑣𝑢𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅))}
248, 23nfcxfr 2175 . . . . . . . . . 10 𝑥𝐹
25 nfcv 2178 . . . . . . . . . 10 𝑥𝑦
2619, 24, 25nfov 5535 . . . . . . . . 9 𝑥(⟨𝐴, 𝐵𝐹𝑦)
2726nfeq1 2187 . . . . . . . 8 𝑥(⟨𝐴, 𝐵𝐹𝑦) = 𝑧
2822, 27nfim 1464 . . . . . . 7 𝑥(∃𝑤𝑣𝑢𝑓((⟨𝐴, 𝐵⟩ = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅) → (⟨𝐴, 𝐵𝐹𝑦) = 𝑧)
29 nfv 1421 . . . . . . . 8 𝑦𝑤𝑣𝑢𝑓((⟨𝐴, 𝐵⟩ = ⟨𝑤, 𝑣⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅)
30 nfoprab2 5555 . . . . . . . . . . 11 𝑦{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ (𝐻 × 𝐻) ∧ 𝑦 ∈ (𝐻 × 𝐻)) ∧ ∃𝑤𝑣𝑢𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅))}
318, 30nfcxfr 2175 . . . . . . . . . 10 𝑦𝐹
3220, 31, 21nfov 5535 . . . . . . . . 9 𝑦(⟨𝐴, 𝐵𝐹𝐶, 𝐷⟩)
3332nfeq1 2187 . . . . . . . 8 𝑦(⟨𝐴, 𝐵𝐹𝐶, 𝐷⟩) = 𝑧
3429, 33nfim 1464 . . . . . . 7 𝑦(∃𝑤𝑣𝑢𝑓((⟨𝐴, 𝐵⟩ = ⟨𝑤, 𝑣⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅) → (⟨𝐴, 𝐵𝐹𝐶, 𝐷⟩) = 𝑧)
35 eqeq1 2046 . . . . . . . . . . 11 (𝑥 = ⟨𝐴, 𝐵⟩ → (𝑥 = ⟨𝑤, 𝑣⟩ ↔ ⟨𝐴, 𝐵⟩ = ⟨𝑤, 𝑣⟩))
3635anbi1d 438 . . . . . . . . . 10 (𝑥 = ⟨𝐴, 𝐵⟩ → ((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ↔ (⟨𝐴, 𝐵⟩ = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩)))
3736anbi1d 438 . . . . . . . . 9 (𝑥 = ⟨𝐴, 𝐵⟩ → (((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅) ↔ ((⟨𝐴, 𝐵⟩ = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅)))
38374exbidv 1750 . . . . . . . 8 (𝑥 = ⟨𝐴, 𝐵⟩ → (∃𝑤𝑣𝑢𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅) ↔ ∃𝑤𝑣𝑢𝑓((⟨𝐴, 𝐵⟩ = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅)))
39 oveq1 5519 . . . . . . . . 9 (𝑥 = ⟨𝐴, 𝐵⟩ → (𝑥𝐹𝑦) = (⟨𝐴, 𝐵𝐹𝑦))
4039eqeq1d 2048 . . . . . . . 8 (𝑥 = ⟨𝐴, 𝐵⟩ → ((𝑥𝐹𝑦) = 𝑧 ↔ (⟨𝐴, 𝐵𝐹𝑦) = 𝑧))
4138, 40imbi12d 223 . . . . . . 7 (𝑥 = ⟨𝐴, 𝐵⟩ → ((∃𝑤𝑣𝑢𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅) → (𝑥𝐹𝑦) = 𝑧) ↔ (∃𝑤𝑣𝑢𝑓((⟨𝐴, 𝐵⟩ = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅) → (⟨𝐴, 𝐵𝐹𝑦) = 𝑧)))
42 eqeq1 2046 . . . . . . . . . . 11 (𝑦 = ⟨𝐶, 𝐷⟩ → (𝑦 = ⟨𝑢, 𝑓⟩ ↔ ⟨𝐶, 𝐷⟩ = ⟨𝑢, 𝑓⟩))
4342anbi2d 437 . . . . . . . . . 10 (𝑦 = ⟨𝐶, 𝐷⟩ → ((⟨𝐴, 𝐵⟩ = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ↔ (⟨𝐴, 𝐵⟩ = ⟨𝑤, 𝑣⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑢, 𝑓⟩)))
4443anbi1d 438 . . . . . . . . 9 (𝑦 = ⟨𝐶, 𝐷⟩ → (((⟨𝐴, 𝐵⟩ = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅) ↔ ((⟨𝐴, 𝐵⟩ = ⟨𝑤, 𝑣⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅)))
45444exbidv 1750 . . . . . . . 8 (𝑦 = ⟨𝐶, 𝐷⟩ → (∃𝑤𝑣𝑢𝑓((⟨𝐴, 𝐵⟩ = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅) ↔ ∃𝑤𝑣𝑢𝑓((⟨𝐴, 𝐵⟩ = ⟨𝑤, 𝑣⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅)))
46 oveq2 5520 . . . . . . . . 9 (𝑦 = ⟨𝐶, 𝐷⟩ → (⟨𝐴, 𝐵𝐹𝑦) = (⟨𝐴, 𝐵𝐹𝐶, 𝐷⟩))
4746eqeq1d 2048 . . . . . . . 8 (𝑦 = ⟨𝐶, 𝐷⟩ → ((⟨𝐴, 𝐵𝐹𝑦) = 𝑧 ↔ (⟨𝐴, 𝐵𝐹𝐶, 𝐷⟩) = 𝑧))
4845, 47imbi12d 223 . . . . . . 7 (𝑦 = ⟨𝐶, 𝐷⟩ → ((∃𝑤𝑣𝑢𝑓((⟨𝐴, 𝐵⟩ = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅) → (⟨𝐴, 𝐵𝐹𝑦) = 𝑧) ↔ (∃𝑤𝑣𝑢𝑓((⟨𝐴, 𝐵⟩ = ⟨𝑤, 𝑣⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅) → (⟨𝐴, 𝐵𝐹𝐶, 𝐷⟩) = 𝑧)))
49 moeq 2716 . . . . . . . . . . . 12 ∃*𝑧 𝑧 = 𝑅
5049mosubop 4406 . . . . . . . . . . 11 ∃*𝑧𝑢𝑓(𝑦 = ⟨𝑢, 𝑓⟩ ∧ 𝑧 = 𝑅)
5150mosubop 4406 . . . . . . . . . 10 ∃*𝑧𝑤𝑣(𝑥 = ⟨𝑤, 𝑣⟩ ∧ ∃𝑢𝑓(𝑦 = ⟨𝑢, 𝑓⟩ ∧ 𝑧 = 𝑅))
52 anass 381 . . . . . . . . . . . . . 14 (((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅) ↔ (𝑥 = ⟨𝑤, 𝑣⟩ ∧ (𝑦 = ⟨𝑢, 𝑓⟩ ∧ 𝑧 = 𝑅)))
53522exbii 1497 . . . . . . . . . . . . 13 (∃𝑢𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅) ↔ ∃𝑢𝑓(𝑥 = ⟨𝑤, 𝑣⟩ ∧ (𝑦 = ⟨𝑢, 𝑓⟩ ∧ 𝑧 = 𝑅)))
54 19.42vv 1788 . . . . . . . . . . . . 13 (∃𝑢𝑓(𝑥 = ⟨𝑤, 𝑣⟩ ∧ (𝑦 = ⟨𝑢, 𝑓⟩ ∧ 𝑧 = 𝑅)) ↔ (𝑥 = ⟨𝑤, 𝑣⟩ ∧ ∃𝑢𝑓(𝑦 = ⟨𝑢, 𝑓⟩ ∧ 𝑧 = 𝑅)))
5553, 54bitri 173 . . . . . . . . . . . 12 (∃𝑢𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅) ↔ (𝑥 = ⟨𝑤, 𝑣⟩ ∧ ∃𝑢𝑓(𝑦 = ⟨𝑢, 𝑓⟩ ∧ 𝑧 = 𝑅)))
56552exbii 1497 . . . . . . . . . . 11 (∃𝑤𝑣𝑢𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅) ↔ ∃𝑤𝑣(𝑥 = ⟨𝑤, 𝑣⟩ ∧ ∃𝑢𝑓(𝑦 = ⟨𝑢, 𝑓⟩ ∧ 𝑧 = 𝑅)))
5756mobii 1937 . . . . . . . . . 10 (∃*𝑧𝑤𝑣𝑢𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅) ↔ ∃*𝑧𝑤𝑣(𝑥 = ⟨𝑤, 𝑣⟩ ∧ ∃𝑢𝑓(𝑦 = ⟨𝑢, 𝑓⟩ ∧ 𝑧 = 𝑅)))
5851, 57mpbir 134 . . . . . . . . 9 ∃*𝑧𝑤𝑣𝑢𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅)
5958a1i 9 . . . . . . . 8 ((𝑥 ∈ (𝐻 × 𝐻) ∧ 𝑦 ∈ (𝐻 × 𝐻)) → ∃*𝑧𝑤𝑣𝑢𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅))
6059, 8ovidi 5619 . . . . . . 7 ((𝑥 ∈ (𝐻 × 𝐻) ∧ 𝑦 ∈ (𝐻 × 𝐻)) → (∃𝑤𝑣𝑢𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅) → (𝑥𝐹𝑦) = 𝑧))
6119, 20, 21, 28, 34, 41, 48, 60vtocl2gaf 2620 . . . . . 6 ((⟨𝐴, 𝐵⟩ ∈ (𝐻 × 𝐻) ∧ ⟨𝐶, 𝐷⟩ ∈ (𝐻 × 𝐻)) → (∃𝑤𝑣𝑢𝑓((⟨𝐴, 𝐵⟩ = ⟨𝑤, 𝑣⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅) → (⟨𝐴, 𝐵𝐹𝐶, 𝐷⟩) = 𝑧))
6217, 18, 61syl2an 273 . . . . 5 (((𝐴𝐻𝐵𝐻) ∧ (𝐶𝐻𝐷𝐻)) → (∃𝑤𝑣𝑢𝑓((⟨𝐴, 𝐵⟩ = ⟨𝑤, 𝑣⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅) → (⟨𝐴, 𝐵𝐹𝐶, 𝐷⟩) = 𝑧))
6316, 62sylbird 159 . . . 4 (((𝐴𝐻𝐵𝐻) ∧ (𝐶𝐻𝐷𝐻)) → (𝑧 = 𝑆 → (⟨𝐴, 𝐵𝐹𝐶, 𝐷⟩) = 𝑧))
64 eqeq2 2049 . . . 4 (𝑧 = 𝑆 → ((⟨𝐴, 𝐵𝐹𝐶, 𝐷⟩) = 𝑧 ↔ (⟨𝐴, 𝐵𝐹𝐶, 𝐷⟩) = 𝑆))
6563, 64mpbidi 140 . . 3 (((𝐴𝐻𝐵𝐻) ∧ (𝐶𝐻𝐷𝐻)) → (𝑧 = 𝑆 → (⟨𝐴, 𝐵𝐹𝐶, 𝐷⟩) = 𝑆))
666, 13, 65exlimd 1488 . 2 (((𝐴𝐻𝐵𝐻) ∧ (𝐶𝐻𝐷𝐻)) → (∃𝑧 𝑧 = 𝑆 → (⟨𝐴, 𝐵𝐹𝐶, 𝐷⟩) = 𝑆))
675, 66mpd 13 1 (((𝐴𝐻𝐵𝐻) ∧ (𝐶𝐻𝐷𝐻)) → (⟨𝐴, 𝐵𝐹𝐶, 𝐷⟩) = 𝑆)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   = wceq 1243  ∃wex 1381   ∈ wcel 1393  ∃*wmo 1901  Vcvv 2557  ⟨cop 3378   × cxp 4343  (class class class)co 5512  {coprab 5513 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-setind 4262 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-iota 4867  df-fun 4904  df-fv 4910  df-ov 5515  df-oprab 5516 This theorem is referenced by:  oviec  6212  addcnsr  6910  mulcnsr  6911
 Copyright terms: Public domain W3C validator