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Theorem ov3 6751
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 3200 . 2 𝑧 𝑧 = 𝑆
3 nfv 1845 . . 3 𝑧((𝐴𝐻𝐵𝐻) ∧ (𝐶𝐻𝐷𝐻))
4 nfcv 2767 . . . . 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 6660 . . . . . 6 𝑧{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ (𝐻 × 𝐻) ∧ 𝑦 ∈ (𝐻 × 𝐻)) ∧ ∃𝑤𝑣𝑢𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅))}
75, 6nfcxfr 2765 . . . . 5 𝑧𝐹
8 nfcv 2767 . . . . 5 𝑧𝐶, 𝐷
94, 7, 8nfov 6631 . . . 4 𝑧(⟨𝐴, 𝐵𝐹𝐶, 𝐷⟩)
109nfeq1 2780 . . 3 𝑧(⟨𝐴, 𝐵𝐹𝐶, 𝐷⟩) = 𝑆
11 ov3.2 . . . . . . 7 (((𝑤 = 𝐴𝑣 = 𝐵) ∧ (𝑢 = 𝐶𝑓 = 𝐷)) → 𝑅 = 𝑆)
1211eqeq2d 2636 . . . . . 6 (((𝑤 = 𝐴𝑣 = 𝐵) ∧ (𝑢 = 𝐶𝑓 = 𝐷)) → (𝑧 = 𝑅𝑧 = 𝑆))
1312copsex4g 4924 . . . . 5 (((𝐴𝐻𝐵𝐻) ∧ (𝐶𝐻𝐷𝐻)) → (∃𝑤𝑣𝑢𝑓((⟨𝐴, 𝐵⟩ = ⟨𝑤, 𝑣⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅) ↔ 𝑧 = 𝑆))
14 opelxpi 5113 . . . . . 6 ((𝐴𝐻𝐵𝐻) → ⟨𝐴, 𝐵⟩ ∈ (𝐻 × 𝐻))
15 opelxpi 5113 . . . . . 6 ((𝐶𝐻𝐷𝐻) → ⟨𝐶, 𝐷⟩ ∈ (𝐻 × 𝐻))
16 nfcv 2767 . . . . . . 7 𝑥𝐴, 𝐵
17 nfcv 2767 . . . . . . 7 𝑦𝐴, 𝐵
18 nfcv 2767 . . . . . . 7 𝑦𝐶, 𝐷
19 nfv 1845 . . . . . . . 8 𝑥𝑤𝑣𝑢𝑓((⟨𝐴, 𝐵⟩ = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅)
20 nfoprab1 6658 . . . . . . . . . . 11 𝑥{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ (𝐻 × 𝐻) ∧ 𝑦 ∈ (𝐻 × 𝐻)) ∧ ∃𝑤𝑣𝑢𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅))}
215, 20nfcxfr 2765 . . . . . . . . . 10 𝑥𝐹
22 nfcv 2767 . . . . . . . . . 10 𝑥𝑦
2316, 21, 22nfov 6631 . . . . . . . . 9 𝑥(⟨𝐴, 𝐵𝐹𝑦)
2423nfeq1 2780 . . . . . . . 8 𝑥(⟨𝐴, 𝐵𝐹𝑦) = 𝑧
2519, 24nfim 1827 . . . . . . 7 𝑥(∃𝑤𝑣𝑢𝑓((⟨𝐴, 𝐵⟩ = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅) → (⟨𝐴, 𝐵𝐹𝑦) = 𝑧)
26 nfv 1845 . . . . . . . 8 𝑦𝑤𝑣𝑢𝑓((⟨𝐴, 𝐵⟩ = ⟨𝑤, 𝑣⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅)
27 nfoprab2 6659 . . . . . . . . . . 11 𝑦{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ (𝐻 × 𝐻) ∧ 𝑦 ∈ (𝐻 × 𝐻)) ∧ ∃𝑤𝑣𝑢𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅))}
285, 27nfcxfr 2765 . . . . . . . . . 10 𝑦𝐹
2917, 28, 18nfov 6631 . . . . . . . . 9 𝑦(⟨𝐴, 𝐵𝐹𝐶, 𝐷⟩)
3029nfeq1 2780 . . . . . . . 8 𝑦(⟨𝐴, 𝐵𝐹𝐶, 𝐷⟩) = 𝑧
3126, 30nfim 1827 . . . . . . 7 𝑦(∃𝑤𝑣𝑢𝑓((⟨𝐴, 𝐵⟩ = ⟨𝑤, 𝑣⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅) → (⟨𝐴, 𝐵𝐹𝐶, 𝐷⟩) = 𝑧)
32 eqeq1 2630 . . . . . . . . . . 11 (𝑥 = ⟨𝐴, 𝐵⟩ → (𝑥 = ⟨𝑤, 𝑣⟩ ↔ ⟨𝐴, 𝐵⟩ = ⟨𝑤, 𝑣⟩))
3332anbi1d 740 . . . . . . . . . 10 (𝑥 = ⟨𝐴, 𝐵⟩ → ((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ↔ (⟨𝐴, 𝐵⟩ = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩)))
3433anbi1d 740 . . . . . . . . 9 (𝑥 = ⟨𝐴, 𝐵⟩ → (((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅) ↔ ((⟨𝐴, 𝐵⟩ = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅)))
35344exbidv 1856 . . . . . . . 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 6612 . . . . . . . . 9 (𝑥 = ⟨𝐴, 𝐵⟩ → (𝑥𝐹𝑦) = (⟨𝐴, 𝐵𝐹𝑦))
3736eqeq1d 2628 . . . . . . . 8 (𝑥 = ⟨𝐴, 𝐵⟩ → ((𝑥𝐹𝑦) = 𝑧 ↔ (⟨𝐴, 𝐵𝐹𝑦) = 𝑧))
3835, 37imbi12d 334 . . . . . . 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 2630 . . . . . . . . . . 11 (𝑦 = ⟨𝐶, 𝐷⟩ → (𝑦 = ⟨𝑢, 𝑓⟩ ↔ ⟨𝐶, 𝐷⟩ = ⟨𝑢, 𝑓⟩))
4039anbi2d 739 . . . . . . . . . 10 (𝑦 = ⟨𝐶, 𝐷⟩ → ((⟨𝐴, 𝐵⟩ = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ↔ (⟨𝐴, 𝐵⟩ = ⟨𝑤, 𝑣⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑢, 𝑓⟩)))
4140anbi1d 740 . . . . . . . . 9 (𝑦 = ⟨𝐶, 𝐷⟩ → (((⟨𝐴, 𝐵⟩ = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅) ↔ ((⟨𝐴, 𝐵⟩ = ⟨𝑤, 𝑣⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅)))
42414exbidv 1856 . . . . . . . 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 6613 . . . . . . . . 9 (𝑦 = ⟨𝐶, 𝐷⟩ → (⟨𝐴, 𝐵𝐹𝑦) = (⟨𝐴, 𝐵𝐹𝐶, 𝐷⟩))
4443eqeq1d 2628 . . . . . . . 8 (𝑦 = ⟨𝐶, 𝐷⟩ → ((⟨𝐴, 𝐵𝐹𝑦) = 𝑧 ↔ (⟨𝐴, 𝐵𝐹𝐶, 𝐷⟩) = 𝑧))
4542, 44imbi12d 334 . . . . . . 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 3369 . . . . . . . . . . . 12 ∃*𝑧 𝑧 = 𝑅
4746mosubop 4938 . . . . . . . . . . 11 ∃*𝑧𝑢𝑓(𝑦 = ⟨𝑢, 𝑓⟩ ∧ 𝑧 = 𝑅)
4847mosubop 4938 . . . . . . . . . 10 ∃*𝑧𝑤𝑣(𝑥 = ⟨𝑤, 𝑣⟩ ∧ ∃𝑢𝑓(𝑦 = ⟨𝑢, 𝑓⟩ ∧ 𝑧 = 𝑅))
49 anass 680 . . . . . . . . . . . . . 14 (((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅) ↔ (𝑥 = ⟨𝑤, 𝑣⟩ ∧ (𝑦 = ⟨𝑢, 𝑓⟩ ∧ 𝑧 = 𝑅)))
50492exbii 1773 . . . . . . . . . . . . 13 (∃𝑢𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅) ↔ ∃𝑢𝑓(𝑥 = ⟨𝑤, 𝑣⟩ ∧ (𝑦 = ⟨𝑢, 𝑓⟩ ∧ 𝑧 = 𝑅)))
51 19.42vv 1922 . . . . . . . . . . . . 13 (∃𝑢𝑓(𝑥 = ⟨𝑤, 𝑣⟩ ∧ (𝑦 = ⟨𝑢, 𝑓⟩ ∧ 𝑧 = 𝑅)) ↔ (𝑥 = ⟨𝑤, 𝑣⟩ ∧ ∃𝑢𝑓(𝑦 = ⟨𝑢, 𝑓⟩ ∧ 𝑧 = 𝑅)))
5250, 51bitri 264 . . . . . . . . . . . 12 (∃𝑢𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅) ↔ (𝑥 = ⟨𝑤, 𝑣⟩ ∧ ∃𝑢𝑓(𝑦 = ⟨𝑢, 𝑓⟩ ∧ 𝑧 = 𝑅)))
53522exbii 1773 . . . . . . . . . . 11 (∃𝑤𝑣𝑢𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅) ↔ ∃𝑤𝑣(𝑥 = ⟨𝑤, 𝑣⟩ ∧ ∃𝑢𝑓(𝑦 = ⟨𝑢, 𝑓⟩ ∧ 𝑧 = 𝑅)))
5453mobii 2497 . . . . . . . . . 10 (∃*𝑧𝑤𝑣𝑢𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅) ↔ ∃*𝑧𝑤𝑣(𝑥 = ⟨𝑤, 𝑣⟩ ∧ ∃𝑢𝑓(𝑦 = ⟨𝑢, 𝑓⟩ ∧ 𝑧 = 𝑅)))
5548, 54mpbir 221 . . . . . . . . 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 6733 . . . . . . 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 3264 . . . . . 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 494 . . . . 5 (((𝐴𝐻𝐵𝐻) ∧ (𝐶𝐻𝐷𝐻)) → (∃𝑤𝑣𝑢𝑓((⟨𝐴, 𝐵⟩ = ⟨𝑤, 𝑣⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅) → (⟨𝐴, 𝐵𝐹𝐶, 𝐷⟩) = 𝑧))
6013, 59sylbird 250 . . . 4 (((𝐴𝐻𝐵𝐻) ∧ (𝐶𝐻𝐷𝐻)) → (𝑧 = 𝑆 → (⟨𝐴, 𝐵𝐹𝐶, 𝐷⟩) = 𝑧))
61 eqeq2 2637 . . . 4 (𝑧 = 𝑆 → ((⟨𝐴, 𝐵𝐹𝐶, 𝐷⟩) = 𝑧 ↔ (⟨𝐴, 𝐵𝐹𝐶, 𝐷⟩) = 𝑆))
6260, 61mpbidi 231 . . 3 (((𝐴𝐻𝐵𝐻) ∧ (𝐶𝐻𝐷𝐻)) → (𝑧 = 𝑆 → (⟨𝐴, 𝐵𝐹𝐶, 𝐷⟩) = 𝑆))
633, 10, 62exlimd 2090 . 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 384   = wceq 1480  wex 1701  wcel 1992  ∃*wmo 2475  Vcvv 3191  cop 4159   × cxp 5077  (class class class)co 6605  {coprab 6606
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-iota 5813  df-fun 5852  df-fv 5858  df-ov 6608  df-oprab 6609
This theorem is referenced by:  addcnsr  9901  mulcnsr  9902
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