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Theorem ov3 7313
 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
ov3.1 𝑆 ∈ V
ov3.2 (((𝑤 = 𝐴𝑣 = 𝐵) ∧ (𝑢 = 𝐶𝑓 = 𝐷)) → 𝑅 = 𝑆)
ov3.3 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ (𝐻 × 𝐻) ∧ 𝑦 ∈ (𝐻 × 𝐻)) ∧ ∃𝑤𝑣𝑢𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅))}
Assertion
Ref Expression
ov3 (((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 ov3
StepHypRef Expression
1 ov3.1 . . 3 𝑆 ∈ V
21isseti 3424 . 2 𝑧 𝑧 = 𝑆
3 nfv 1915 . . 3 𝑧((𝐴𝐻𝐵𝐻) ∧ (𝐶𝐻𝐷𝐻))
4 nfcv 2919 . . . . 5 𝑧𝐴, 𝐵
5 ov3.3 . . . . . 6 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ (𝐻 × 𝐻) ∧ 𝑦 ∈ (𝐻 × 𝐻)) ∧ ∃𝑤𝑣𝑢𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅))}
6 nfoprab3 7217 . . . . . 6 𝑧{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ (𝐻 × 𝐻) ∧ 𝑦 ∈ (𝐻 × 𝐻)) ∧ ∃𝑤𝑣𝑢𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅))}
75, 6nfcxfr 2917 . . . . 5 𝑧𝐹
8 nfcv 2919 . . . . 5 𝑧𝐶, 𝐷
94, 7, 8nfov 7186 . . . 4 𝑧(⟨𝐴, 𝐵𝐹𝐶, 𝐷⟩)
109nfeq1 2934 . . 3 𝑧(⟨𝐴, 𝐵𝐹𝐶, 𝐷⟩) = 𝑆
11 ov3.2 . . . . . . 7 (((𝑤 = 𝐴𝑣 = 𝐵) ∧ (𝑢 = 𝐶𝑓 = 𝐷)) → 𝑅 = 𝑆)
1211eqeq2d 2769 . . . . . 6 (((𝑤 = 𝐴𝑣 = 𝐵) ∧ (𝑢 = 𝐶𝑓 = 𝐷)) → (𝑧 = 𝑅𝑧 = 𝑆))
1312copsex4g 5358 . . . . 5 (((𝐴𝐻𝐵𝐻) ∧ (𝐶𝐻𝐷𝐻)) → (∃𝑤𝑣𝑢𝑓((⟨𝐴, 𝐵⟩ = ⟨𝑤, 𝑣⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅) ↔ 𝑧 = 𝑆))
14 opelxpi 5565 . . . . . 6 ((𝐴𝐻𝐵𝐻) → ⟨𝐴, 𝐵⟩ ∈ (𝐻 × 𝐻))
15 opelxpi 5565 . . . . . 6 ((𝐶𝐻𝐷𝐻) → ⟨𝐶, 𝐷⟩ ∈ (𝐻 × 𝐻))
16 nfcv 2919 . . . . . . 7 𝑥𝐴, 𝐵
17 nfcv 2919 . . . . . . 7 𝑦𝐴, 𝐵
18 nfcv 2919 . . . . . . 7 𝑦𝐶, 𝐷
19 nfv 1915 . . . . . . . 8 𝑥𝑤𝑣𝑢𝑓((⟨𝐴, 𝐵⟩ = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅)
20 nfoprab1 7215 . . . . . . . . . . 11 𝑥{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ (𝐻 × 𝐻) ∧ 𝑦 ∈ (𝐻 × 𝐻)) ∧ ∃𝑤𝑣𝑢𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅))}
215, 20nfcxfr 2917 . . . . . . . . . 10 𝑥𝐹
22 nfcv 2919 . . . . . . . . . 10 𝑥𝑦
2316, 21, 22nfov 7186 . . . . . . . . 9 𝑥(⟨𝐴, 𝐵𝐹𝑦)
2423nfeq1 2934 . . . . . . . 8 𝑥(⟨𝐴, 𝐵𝐹𝑦) = 𝑧
2519, 24nfim 1897 . . . . . . 7 𝑥(∃𝑤𝑣𝑢𝑓((⟨𝐴, 𝐵⟩ = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅) → (⟨𝐴, 𝐵𝐹𝑦) = 𝑧)
26 nfv 1915 . . . . . . . 8 𝑦𝑤𝑣𝑢𝑓((⟨𝐴, 𝐵⟩ = ⟨𝑤, 𝑣⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅)
27 nfoprab2 7216 . . . . . . . . . . 11 𝑦{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ (𝐻 × 𝐻) ∧ 𝑦 ∈ (𝐻 × 𝐻)) ∧ ∃𝑤𝑣𝑢𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅))}
285, 27nfcxfr 2917 . . . . . . . . . 10 𝑦𝐹
2917, 28, 18nfov 7186 . . . . . . . . 9 𝑦(⟨𝐴, 𝐵𝐹𝐶, 𝐷⟩)
3029nfeq1 2934 . . . . . . . 8 𝑦(⟨𝐴, 𝐵𝐹𝐶, 𝐷⟩) = 𝑧
3126, 30nfim 1897 . . . . . . 7 𝑦(∃𝑤𝑣𝑢𝑓((⟨𝐴, 𝐵⟩ = ⟨𝑤, 𝑣⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅) → (⟨𝐴, 𝐵𝐹𝐶, 𝐷⟩) = 𝑧)
32 eqeq1 2762 . . . . . . . . . . 11 (𝑥 = ⟨𝐴, 𝐵⟩ → (𝑥 = ⟨𝑤, 𝑣⟩ ↔ ⟨𝐴, 𝐵⟩ = ⟨𝑤, 𝑣⟩))
3332anbi1d 632 . . . . . . . . . 10 (𝑥 = ⟨𝐴, 𝐵⟩ → ((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ↔ (⟨𝐴, 𝐵⟩ = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩)))
3433anbi1d 632 . . . . . . . . 9 (𝑥 = ⟨𝐴, 𝐵⟩ → (((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅) ↔ ((⟨𝐴, 𝐵⟩ = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅)))
35344exbidv 1927 . . . . . . . 8 (𝑥 = ⟨𝐴, 𝐵⟩ → (∃𝑤𝑣𝑢𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅) ↔ ∃𝑤𝑣𝑢𝑓((⟨𝐴, 𝐵⟩ = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅)))
36 oveq1 7163 . . . . . . . . 9 (𝑥 = ⟨𝐴, 𝐵⟩ → (𝑥𝐹𝑦) = (⟨𝐴, 𝐵𝐹𝑦))
3736eqeq1d 2760 . . . . . . . 8 (𝑥 = ⟨𝐴, 𝐵⟩ → ((𝑥𝐹𝑦) = 𝑧 ↔ (⟨𝐴, 𝐵𝐹𝑦) = 𝑧))
3835, 37imbi12d 348 . . . . . . 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)))
39 eqeq1 2762 . . . . . . . . . . 11 (𝑦 = ⟨𝐶, 𝐷⟩ → (𝑦 = ⟨𝑢, 𝑓⟩ ↔ ⟨𝐶, 𝐷⟩ = ⟨𝑢, 𝑓⟩))
4039anbi2d 631 . . . . . . . . . 10 (𝑦 = ⟨𝐶, 𝐷⟩ → ((⟨𝐴, 𝐵⟩ = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ↔ (⟨𝐴, 𝐵⟩ = ⟨𝑤, 𝑣⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑢, 𝑓⟩)))
4140anbi1d 632 . . . . . . . . 9 (𝑦 = ⟨𝐶, 𝐷⟩ → (((⟨𝐴, 𝐵⟩ = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅) ↔ ((⟨𝐴, 𝐵⟩ = ⟨𝑤, 𝑣⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅)))
42414exbidv 1927 . . . . . . . 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)))
43 oveq2 7164 . . . . . . . . 9 (𝑦 = ⟨𝐶, 𝐷⟩ → (⟨𝐴, 𝐵𝐹𝑦) = (⟨𝐴, 𝐵𝐹𝐶, 𝐷⟩))
4443eqeq1d 2760 . . . . . . . 8 (𝑦 = ⟨𝐶, 𝐷⟩ → ((⟨𝐴, 𝐵𝐹𝑦) = 𝑧 ↔ (⟨𝐴, 𝐵𝐹𝐶, 𝐷⟩) = 𝑧))
4542, 44imbi12d 348 . . . . . . 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)))
46 moeq 3623 . . . . . . . . . . . 12 ∃*𝑧 𝑧 = 𝑅
4746mosubop 5374 . . . . . . . . . . 11 ∃*𝑧𝑢𝑓(𝑦 = ⟨𝑢, 𝑓⟩ ∧ 𝑧 = 𝑅)
4847mosubop 5374 . . . . . . . . . 10 ∃*𝑧𝑤𝑣(𝑥 = ⟨𝑤, 𝑣⟩ ∧ ∃𝑢𝑓(𝑦 = ⟨𝑢, 𝑓⟩ ∧ 𝑧 = 𝑅))
49 anass 472 . . . . . . . . . . . . . 14 (((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅) ↔ (𝑥 = ⟨𝑤, 𝑣⟩ ∧ (𝑦 = ⟨𝑢, 𝑓⟩ ∧ 𝑧 = 𝑅)))
50492exbii 1850 . . . . . . . . . . . . 13 (∃𝑢𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅) ↔ ∃𝑢𝑓(𝑥 = ⟨𝑤, 𝑣⟩ ∧ (𝑦 = ⟨𝑢, 𝑓⟩ ∧ 𝑧 = 𝑅)))
51 19.42vv 1958 . . . . . . . . . . . . 13 (∃𝑢𝑓(𝑥 = ⟨𝑤, 𝑣⟩ ∧ (𝑦 = ⟨𝑢, 𝑓⟩ ∧ 𝑧 = 𝑅)) ↔ (𝑥 = ⟨𝑤, 𝑣⟩ ∧ ∃𝑢𝑓(𝑦 = ⟨𝑢, 𝑓⟩ ∧ 𝑧 = 𝑅)))
5250, 51bitri 278 . . . . . . . . . . . 12 (∃𝑢𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅) ↔ (𝑥 = ⟨𝑤, 𝑣⟩ ∧ ∃𝑢𝑓(𝑦 = ⟨𝑢, 𝑓⟩ ∧ 𝑧 = 𝑅)))
53522exbii 1850 . . . . . . . . . . 11 (∃𝑤𝑣𝑢𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅) ↔ ∃𝑤𝑣(𝑥 = ⟨𝑤, 𝑣⟩ ∧ ∃𝑢𝑓(𝑦 = ⟨𝑢, 𝑓⟩ ∧ 𝑧 = 𝑅)))
5453mobii 2565 . . . . . . . . . 10 (∃*𝑧𝑤𝑣𝑢𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅) ↔ ∃*𝑧𝑤𝑣(𝑥 = ⟨𝑤, 𝑣⟩ ∧ ∃𝑢𝑓(𝑦 = ⟨𝑢, 𝑓⟩ ∧ 𝑧 = 𝑅)))
5548, 54mpbir 234 . . . . . . . . 9 ∃*𝑧𝑤𝑣𝑢𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅)
5655a1i 11 . . . . . . . 8 ((𝑥 ∈ (𝐻 × 𝐻) ∧ 𝑦 ∈ (𝐻 × 𝐻)) → ∃*𝑧𝑤𝑣𝑢𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅))
5756, 5ovidi 7294 . . . . . . 7 ((𝑥 ∈ (𝐻 × 𝐻) ∧ 𝑦 ∈ (𝐻 × 𝐻)) → (∃𝑤𝑣𝑢𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅) → (𝑥𝐹𝑦) = 𝑧))
5816, 17, 18, 25, 31, 38, 45, 57vtocl2gaf 3496 . . . . . 6 ((⟨𝐴, 𝐵⟩ ∈ (𝐻 × 𝐻) ∧ ⟨𝐶, 𝐷⟩ ∈ (𝐻 × 𝐻)) → (∃𝑤𝑣𝑢𝑓((⟨𝐴, 𝐵⟩ = ⟨𝑤, 𝑣⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅) → (⟨𝐴, 𝐵𝐹𝐶, 𝐷⟩) = 𝑧))
5914, 15, 58syl2an 598 . . . . 5 (((𝐴𝐻𝐵𝐻) ∧ (𝐶𝐻𝐷𝐻)) → (∃𝑤𝑣𝑢𝑓((⟨𝐴, 𝐵⟩ = ⟨𝑤, 𝑣⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅) → (⟨𝐴, 𝐵𝐹𝐶, 𝐷⟩) = 𝑧))
6013, 59sylbird 263 . . . 4 (((𝐴𝐻𝐵𝐻) ∧ (𝐶𝐻𝐷𝐻)) → (𝑧 = 𝑆 → (⟨𝐴, 𝐵𝐹𝐶, 𝐷⟩) = 𝑧))
61 eqeq2 2770 . . . 4 (𝑧 = 𝑆 → ((⟨𝐴, 𝐵𝐹𝐶, 𝐷⟩) = 𝑧 ↔ (⟨𝐴, 𝐵𝐹𝐶, 𝐷⟩) = 𝑆))
6260, 61mpbidi 244 . . 3 (((𝐴𝐻𝐵𝐻) ∧ (𝐶𝐻𝐷𝐻)) → (𝑧 = 𝑆 → (⟨𝐴, 𝐵𝐹𝐶, 𝐷⟩) = 𝑆))
633, 10, 62exlimd 2216 . 2 (((𝐴𝐻𝐵𝐻) ∧ (𝐶𝐻𝐷𝐻)) → (∃𝑧 𝑧 = 𝑆 → (⟨𝐴, 𝐵𝐹𝐶, 𝐷⟩) = 𝑆))
642, 63mpi 20 1 (((𝐴𝐻𝐵𝐻) ∧ (𝐶𝐻𝐷𝐻)) → (⟨𝐴, 𝐵𝐹𝐶, 𝐷⟩) = 𝑆)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538  ∃wex 1781   ∈ wcel 2111  ∃*wmo 2555  Vcvv 3409  ⟨cop 4531   × cxp 5526  (class class class)co 7156  {coprab 7157 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5173  ax-nul 5180  ax-pr 5302 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ral 3075  df-rex 3076  df-v 3411  df-sbc 3699  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5037  df-opab 5099  df-id 5434  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-iota 6299  df-fun 6342  df-fv 6348  df-ov 7159  df-oprab 7160 This theorem is referenced by:  addcnsr  10608  mulcnsr  10609
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