Metamath Proof Explorer


Theorem ov3

Description: The value of an operation class abstraction. Special case. (Contributed by NM, 28-May-1995) (Revised by Mario Carneiro, 29-Dec-2014)

Ref Expression
Hypotheses 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 ov3 ( ( ( 𝐴𝐻𝐵𝐻 ) ∧ ( 𝐶𝐻𝐷𝐻 ) ) → ( ⟨ 𝐴 , 𝐵𝐹𝐶 , 𝐷 ⟩ ) = 𝑆 )

Proof

Step Hyp Ref Expression
1 ov3.1 𝑆 ∈ V
2 ov3.2 ( ( ( 𝑤 = 𝐴𝑣 = 𝐵 ) ∧ ( 𝑢 = 𝐶𝑓 = 𝐷 ) ) → 𝑅 = 𝑆 )
3 ov3.3 𝐹 = { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ ( ( 𝑥 ∈ ( 𝐻 × 𝐻 ) ∧ 𝑦 ∈ ( 𝐻 × 𝐻 ) ) ∧ ∃ 𝑤𝑣𝑢𝑓 ( ( 𝑥 = ⟨ 𝑤 , 𝑣 ⟩ ∧ 𝑦 = ⟨ 𝑢 , 𝑓 ⟩ ) ∧ 𝑧 = 𝑅 ) ) }
4 1 isseti 𝑧 𝑧 = 𝑆
5 nfv 𝑧 ( ( 𝐴𝐻𝐵𝐻 ) ∧ ( 𝐶𝐻𝐷𝐻 ) )
6 nfcv 𝑧𝐴 , 𝐵
7 nfoprab3 𝑧 { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ ( ( 𝑥 ∈ ( 𝐻 × 𝐻 ) ∧ 𝑦 ∈ ( 𝐻 × 𝐻 ) ) ∧ ∃ 𝑤𝑣𝑢𝑓 ( ( 𝑥 = ⟨ 𝑤 , 𝑣 ⟩ ∧ 𝑦 = ⟨ 𝑢 , 𝑓 ⟩ ) ∧ 𝑧 = 𝑅 ) ) }
8 3 7 nfcxfr 𝑧 𝐹
9 nfcv 𝑧𝐶 , 𝐷
10 6 8 9 nfov 𝑧 ( ⟨ 𝐴 , 𝐵𝐹𝐶 , 𝐷 ⟩ )
11 10 nfeq1 𝑧 ( ⟨ 𝐴 , 𝐵𝐹𝐶 , 𝐷 ⟩ ) = 𝑆
12 2 eqeq2d ( ( ( 𝑤 = 𝐴𝑣 = 𝐵 ) ∧ ( 𝑢 = 𝐶𝑓 = 𝐷 ) ) → ( 𝑧 = 𝑅𝑧 = 𝑆 ) )
13 12 copsex4g ( ( ( 𝐴𝐻𝐵𝐻 ) ∧ ( 𝐶𝐻𝐷𝐻 ) ) → ( ∃ 𝑤𝑣𝑢𝑓 ( ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑤 , 𝑣 ⟩ ∧ ⟨ 𝐶 , 𝐷 ⟩ = ⟨ 𝑢 , 𝑓 ⟩ ) ∧ 𝑧 = 𝑅 ) ↔ 𝑧 = 𝑆 ) )
14 opelxpi ( ( 𝐴𝐻𝐵𝐻 ) → ⟨ 𝐴 , 𝐵 ⟩ ∈ ( 𝐻 × 𝐻 ) )
15 opelxpi ( ( 𝐶𝐻𝐷𝐻 ) → ⟨ 𝐶 , 𝐷 ⟩ ∈ ( 𝐻 × 𝐻 ) )
16 nfcv 𝑥𝐴 , 𝐵
17 nfcv 𝑦𝐴 , 𝐵
18 nfcv 𝑦𝐶 , 𝐷
19 nfv 𝑥𝑤𝑣𝑢𝑓 ( ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑤 , 𝑣 ⟩ ∧ 𝑦 = ⟨ 𝑢 , 𝑓 ⟩ ) ∧ 𝑧 = 𝑅 )
20 nfoprab1 𝑥 { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ ( ( 𝑥 ∈ ( 𝐻 × 𝐻 ) ∧ 𝑦 ∈ ( 𝐻 × 𝐻 ) ) ∧ ∃ 𝑤𝑣𝑢𝑓 ( ( 𝑥 = ⟨ 𝑤 , 𝑣 ⟩ ∧ 𝑦 = ⟨ 𝑢 , 𝑓 ⟩ ) ∧ 𝑧 = 𝑅 ) ) }
21 3 20 nfcxfr 𝑥 𝐹
22 nfcv 𝑥 𝑦
23 16 21 22 nfov 𝑥 ( ⟨ 𝐴 , 𝐵𝐹 𝑦 )
24 23 nfeq1 𝑥 ( ⟨ 𝐴 , 𝐵𝐹 𝑦 ) = 𝑧
25 19 24 nfim 𝑥 ( ∃ 𝑤𝑣𝑢𝑓 ( ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑤 , 𝑣 ⟩ ∧ 𝑦 = ⟨ 𝑢 , 𝑓 ⟩ ) ∧ 𝑧 = 𝑅 ) → ( ⟨ 𝐴 , 𝐵𝐹 𝑦 ) = 𝑧 )
26 nfv 𝑦𝑤𝑣𝑢𝑓 ( ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑤 , 𝑣 ⟩ ∧ ⟨ 𝐶 , 𝐷 ⟩ = ⟨ 𝑢 , 𝑓 ⟩ ) ∧ 𝑧 = 𝑅 )
27 nfoprab2 𝑦 { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ ( ( 𝑥 ∈ ( 𝐻 × 𝐻 ) ∧ 𝑦 ∈ ( 𝐻 × 𝐻 ) ) ∧ ∃ 𝑤𝑣𝑢𝑓 ( ( 𝑥 = ⟨ 𝑤 , 𝑣 ⟩ ∧ 𝑦 = ⟨ 𝑢 , 𝑓 ⟩ ) ∧ 𝑧 = 𝑅 ) ) }
28 3 27 nfcxfr 𝑦 𝐹
29 17 28 18 nfov 𝑦 ( ⟨ 𝐴 , 𝐵𝐹𝐶 , 𝐷 ⟩ )
30 29 nfeq1 𝑦 ( ⟨ 𝐴 , 𝐵𝐹𝐶 , 𝐷 ⟩ ) = 𝑧
31 26 30 nfim 𝑦 ( ∃ 𝑤𝑣𝑢𝑓 ( ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑤 , 𝑣 ⟩ ∧ ⟨ 𝐶 , 𝐷 ⟩ = ⟨ 𝑢 , 𝑓 ⟩ ) ∧ 𝑧 = 𝑅 ) → ( ⟨ 𝐴 , 𝐵𝐹𝐶 , 𝐷 ⟩ ) = 𝑧 )
32 eqeq1 ( 𝑥 = ⟨ 𝐴 , 𝐵 ⟩ → ( 𝑥 = ⟨ 𝑤 , 𝑣 ⟩ ↔ ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑤 , 𝑣 ⟩ ) )
33 32 anbi1d ( 𝑥 = ⟨ 𝐴 , 𝐵 ⟩ → ( ( 𝑥 = ⟨ 𝑤 , 𝑣 ⟩ ∧ 𝑦 = ⟨ 𝑢 , 𝑓 ⟩ ) ↔ ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑤 , 𝑣 ⟩ ∧ 𝑦 = ⟨ 𝑢 , 𝑓 ⟩ ) ) )
34 33 anbi1d ( 𝑥 = ⟨ 𝐴 , 𝐵 ⟩ → ( ( ( 𝑥 = ⟨ 𝑤 , 𝑣 ⟩ ∧ 𝑦 = ⟨ 𝑢 , 𝑓 ⟩ ) ∧ 𝑧 = 𝑅 ) ↔ ( ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑤 , 𝑣 ⟩ ∧ 𝑦 = ⟨ 𝑢 , 𝑓 ⟩ ) ∧ 𝑧 = 𝑅 ) ) )
35 34 4exbidv ( 𝑥 = ⟨ 𝐴 , 𝐵 ⟩ → ( ∃ 𝑤𝑣𝑢𝑓 ( ( 𝑥 = ⟨ 𝑤 , 𝑣 ⟩ ∧ 𝑦 = ⟨ 𝑢 , 𝑓 ⟩ ) ∧ 𝑧 = 𝑅 ) ↔ ∃ 𝑤𝑣𝑢𝑓 ( ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑤 , 𝑣 ⟩ ∧ 𝑦 = ⟨ 𝑢 , 𝑓 ⟩ ) ∧ 𝑧 = 𝑅 ) ) )
36 oveq1 ( 𝑥 = ⟨ 𝐴 , 𝐵 ⟩ → ( 𝑥 𝐹 𝑦 ) = ( ⟨ 𝐴 , 𝐵𝐹 𝑦 ) )
37 36 eqeq1d ( 𝑥 = ⟨ 𝐴 , 𝐵 ⟩ → ( ( 𝑥 𝐹 𝑦 ) = 𝑧 ↔ ( ⟨ 𝐴 , 𝐵𝐹 𝑦 ) = 𝑧 ) )
38 35 37 imbi12d ( 𝑥 = ⟨ 𝐴 , 𝐵 ⟩ → ( ( ∃ 𝑤𝑣𝑢𝑓 ( ( 𝑥 = ⟨ 𝑤 , 𝑣 ⟩ ∧ 𝑦 = ⟨ 𝑢 , 𝑓 ⟩ ) ∧ 𝑧 = 𝑅 ) → ( 𝑥 𝐹 𝑦 ) = 𝑧 ) ↔ ( ∃ 𝑤𝑣𝑢𝑓 ( ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑤 , 𝑣 ⟩ ∧ 𝑦 = ⟨ 𝑢 , 𝑓 ⟩ ) ∧ 𝑧 = 𝑅 ) → ( ⟨ 𝐴 , 𝐵𝐹 𝑦 ) = 𝑧 ) ) )
39 eqeq1 ( 𝑦 = ⟨ 𝐶 , 𝐷 ⟩ → ( 𝑦 = ⟨ 𝑢 , 𝑓 ⟩ ↔ ⟨ 𝐶 , 𝐷 ⟩ = ⟨ 𝑢 , 𝑓 ⟩ ) )
40 39 anbi2d ( 𝑦 = ⟨ 𝐶 , 𝐷 ⟩ → ( ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑤 , 𝑣 ⟩ ∧ 𝑦 = ⟨ 𝑢 , 𝑓 ⟩ ) ↔ ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑤 , 𝑣 ⟩ ∧ ⟨ 𝐶 , 𝐷 ⟩ = ⟨ 𝑢 , 𝑓 ⟩ ) ) )
41 40 anbi1d ( 𝑦 = ⟨ 𝐶 , 𝐷 ⟩ → ( ( ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑤 , 𝑣 ⟩ ∧ 𝑦 = ⟨ 𝑢 , 𝑓 ⟩ ) ∧ 𝑧 = 𝑅 ) ↔ ( ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑤 , 𝑣 ⟩ ∧ ⟨ 𝐶 , 𝐷 ⟩ = ⟨ 𝑢 , 𝑓 ⟩ ) ∧ 𝑧 = 𝑅 ) ) )
42 41 4exbidv ( 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 ( 𝑦 = ⟨ 𝐶 , 𝐷 ⟩ → ( ⟨ 𝐴 , 𝐵𝐹 𝑦 ) = ( ⟨ 𝐴 , 𝐵𝐹𝐶 , 𝐷 ⟩ ) )
44 43 eqeq1d ( 𝑦 = ⟨ 𝐶 , 𝐷 ⟩ → ( ( ⟨ 𝐴 , 𝐵𝐹 𝑦 ) = 𝑧 ↔ ( ⟨ 𝐴 , 𝐵𝐹𝐶 , 𝐷 ⟩ ) = 𝑧 ) )
45 42 44 imbi12d ( 𝑦 = ⟨ 𝐶 , 𝐷 ⟩ → ( ( ∃ 𝑤𝑣𝑢𝑓 ( ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑤 , 𝑣 ⟩ ∧ 𝑦 = ⟨ 𝑢 , 𝑓 ⟩ ) ∧ 𝑧 = 𝑅 ) → ( ⟨ 𝐴 , 𝐵𝐹 𝑦 ) = 𝑧 ) ↔ ( ∃ 𝑤𝑣𝑢𝑓 ( ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑤 , 𝑣 ⟩ ∧ ⟨ 𝐶 , 𝐷 ⟩ = ⟨ 𝑢 , 𝑓 ⟩ ) ∧ 𝑧 = 𝑅 ) → ( ⟨ 𝐴 , 𝐵𝐹𝐶 , 𝐷 ⟩ ) = 𝑧 ) ) )
46 moeq ∃* 𝑧 𝑧 = 𝑅
47 46 mosubop ∃* 𝑧𝑢𝑓 ( 𝑦 = ⟨ 𝑢 , 𝑓 ⟩ ∧ 𝑧 = 𝑅 )
48 47 mosubop ∃* 𝑧𝑤𝑣 ( 𝑥 = ⟨ 𝑤 , 𝑣 ⟩ ∧ ∃ 𝑢𝑓 ( 𝑦 = ⟨ 𝑢 , 𝑓 ⟩ ∧ 𝑧 = 𝑅 ) )
49 anass ( ( ( 𝑥 = ⟨ 𝑤 , 𝑣 ⟩ ∧ 𝑦 = ⟨ 𝑢 , 𝑓 ⟩ ) ∧ 𝑧 = 𝑅 ) ↔ ( 𝑥 = ⟨ 𝑤 , 𝑣 ⟩ ∧ ( 𝑦 = ⟨ 𝑢 , 𝑓 ⟩ ∧ 𝑧 = 𝑅 ) ) )
50 49 2exbii ( ∃ 𝑢𝑓 ( ( 𝑥 = ⟨ 𝑤 , 𝑣 ⟩ ∧ 𝑦 = ⟨ 𝑢 , 𝑓 ⟩ ) ∧ 𝑧 = 𝑅 ) ↔ ∃ 𝑢𝑓 ( 𝑥 = ⟨ 𝑤 , 𝑣 ⟩ ∧ ( 𝑦 = ⟨ 𝑢 , 𝑓 ⟩ ∧ 𝑧 = 𝑅 ) ) )
51 19.42vv ( ∃ 𝑢𝑓 ( 𝑥 = ⟨ 𝑤 , 𝑣 ⟩ ∧ ( 𝑦 = ⟨ 𝑢 , 𝑓 ⟩ ∧ 𝑧 = 𝑅 ) ) ↔ ( 𝑥 = ⟨ 𝑤 , 𝑣 ⟩ ∧ ∃ 𝑢𝑓 ( 𝑦 = ⟨ 𝑢 , 𝑓 ⟩ ∧ 𝑧 = 𝑅 ) ) )
52 50 51 bitri ( ∃ 𝑢𝑓 ( ( 𝑥 = ⟨ 𝑤 , 𝑣 ⟩ ∧ 𝑦 = ⟨ 𝑢 , 𝑓 ⟩ ) ∧ 𝑧 = 𝑅 ) ↔ ( 𝑥 = ⟨ 𝑤 , 𝑣 ⟩ ∧ ∃ 𝑢𝑓 ( 𝑦 = ⟨ 𝑢 , 𝑓 ⟩ ∧ 𝑧 = 𝑅 ) ) )
53 52 2exbii ( ∃ 𝑤𝑣𝑢𝑓 ( ( 𝑥 = ⟨ 𝑤 , 𝑣 ⟩ ∧ 𝑦 = ⟨ 𝑢 , 𝑓 ⟩ ) ∧ 𝑧 = 𝑅 ) ↔ ∃ 𝑤𝑣 ( 𝑥 = ⟨ 𝑤 , 𝑣 ⟩ ∧ ∃ 𝑢𝑓 ( 𝑦 = ⟨ 𝑢 , 𝑓 ⟩ ∧ 𝑧 = 𝑅 ) ) )
54 53 mobii ( ∃* 𝑧𝑤𝑣𝑢𝑓 ( ( 𝑥 = ⟨ 𝑤 , 𝑣 ⟩ ∧ 𝑦 = ⟨ 𝑢 , 𝑓 ⟩ ) ∧ 𝑧 = 𝑅 ) ↔ ∃* 𝑧𝑤𝑣 ( 𝑥 = ⟨ 𝑤 , 𝑣 ⟩ ∧ ∃ 𝑢𝑓 ( 𝑦 = ⟨ 𝑢 , 𝑓 ⟩ ∧ 𝑧 = 𝑅 ) ) )
55 48 54 mpbir ∃* 𝑧𝑤𝑣𝑢𝑓 ( ( 𝑥 = ⟨ 𝑤 , 𝑣 ⟩ ∧ 𝑦 = ⟨ 𝑢 , 𝑓 ⟩ ) ∧ 𝑧 = 𝑅 )
56 55 a1i ( ( 𝑥 ∈ ( 𝐻 × 𝐻 ) ∧ 𝑦 ∈ ( 𝐻 × 𝐻 ) ) → ∃* 𝑧𝑤𝑣𝑢𝑓 ( ( 𝑥 = ⟨ 𝑤 , 𝑣 ⟩ ∧ 𝑦 = ⟨ 𝑢 , 𝑓 ⟩ ) ∧ 𝑧 = 𝑅 ) )
57 56 3 ovidi ( ( 𝑥 ∈ ( 𝐻 × 𝐻 ) ∧ 𝑦 ∈ ( 𝐻 × 𝐻 ) ) → ( ∃ 𝑤𝑣𝑢𝑓 ( ( 𝑥 = ⟨ 𝑤 , 𝑣 ⟩ ∧ 𝑦 = ⟨ 𝑢 , 𝑓 ⟩ ) ∧ 𝑧 = 𝑅 ) → ( 𝑥 𝐹 𝑦 ) = 𝑧 ) )
58 16 17 18 25 31 38 45 57 vtocl2gaf ( ( ⟨ 𝐴 , 𝐵 ⟩ ∈ ( 𝐻 × 𝐻 ) ∧ ⟨ 𝐶 , 𝐷 ⟩ ∈ ( 𝐻 × 𝐻 ) ) → ( ∃ 𝑤𝑣𝑢𝑓 ( ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑤 , 𝑣 ⟩ ∧ ⟨ 𝐶 , 𝐷 ⟩ = ⟨ 𝑢 , 𝑓 ⟩ ) ∧ 𝑧 = 𝑅 ) → ( ⟨ 𝐴 , 𝐵𝐹𝐶 , 𝐷 ⟩ ) = 𝑧 ) )
59 14 15 58 syl2an ( ( ( 𝐴𝐻𝐵𝐻 ) ∧ ( 𝐶𝐻𝐷𝐻 ) ) → ( ∃ 𝑤𝑣𝑢𝑓 ( ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑤 , 𝑣 ⟩ ∧ ⟨ 𝐶 , 𝐷 ⟩ = ⟨ 𝑢 , 𝑓 ⟩ ) ∧ 𝑧 = 𝑅 ) → ( ⟨ 𝐴 , 𝐵𝐹𝐶 , 𝐷 ⟩ ) = 𝑧 ) )
60 13 59 sylbird ( ( ( 𝐴𝐻𝐵𝐻 ) ∧ ( 𝐶𝐻𝐷𝐻 ) ) → ( 𝑧 = 𝑆 → ( ⟨ 𝐴 , 𝐵𝐹𝐶 , 𝐷 ⟩ ) = 𝑧 ) )
61 eqeq2 ( 𝑧 = 𝑆 → ( ( ⟨ 𝐴 , 𝐵𝐹𝐶 , 𝐷 ⟩ ) = 𝑧 ↔ ( ⟨ 𝐴 , 𝐵𝐹𝐶 , 𝐷 ⟩ ) = 𝑆 ) )
62 60 61 mpbidi ( ( ( 𝐴𝐻𝐵𝐻 ) ∧ ( 𝐶𝐻𝐷𝐻 ) ) → ( 𝑧 = 𝑆 → ( ⟨ 𝐴 , 𝐵𝐹𝐶 , 𝐷 ⟩ ) = 𝑆 ) )
63 5 11 62 exlimd ( ( ( 𝐴𝐻𝐵𝐻 ) ∧ ( 𝐶𝐻𝐷𝐻 ) ) → ( ∃ 𝑧 𝑧 = 𝑆 → ( ⟨ 𝐴 , 𝐵𝐹𝐶 , 𝐷 ⟩ ) = 𝑆 ) )
64 4 63 mpi ( ( ( 𝐴𝐻𝐵𝐻 ) ∧ ( 𝐶𝐻𝐷𝐻 ) ) → ( ⟨ 𝐴 , 𝐵𝐹𝐶 , 𝐷 ⟩ ) = 𝑆 )