Metamath Proof Explorer

Theorem ov

Description: The value of an operation class abstraction. (Contributed by NM, 16-May-1995) (Revised by David Abernethy, 19-Jun-2012)

Ref Expression
Hypotheses ov.1 ⊢ 𝐶 ∈ V
ov.2 ⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
ov.3 ⊢ ( 𝑦 = 𝐵 → ( 𝜓 ↔ 𝜒 ) )
ov.4 ⊢ ( 𝑧 = 𝐶 → ( 𝜒 ↔ 𝜃 ) )
ov.5 ⊢ ( ( 𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆 ) → ∃! 𝑧 𝜑 )
ov.6 ⊢ 𝐹 = { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ ( ( 𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆 ) ∧ 𝜑 ) }
Assertion ov ( ( 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ) → ( ( 𝐴 𝐹 𝐵 ) = 𝐶 ↔ 𝜃 ) )

Proof

Step Hyp Ref Expression
1 ov.1 ⊢ 𝐶 ∈ V
2 ov.2 ⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
3 ov.3 ⊢ ( 𝑦 = 𝐵 → ( 𝜓 ↔ 𝜒 ) )
4 ov.4 ⊢ ( 𝑧 = 𝐶 → ( 𝜒 ↔ 𝜃 ) )
5 ov.5 ⊢ ( ( 𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆 ) → ∃! 𝑧 𝜑 )
6 ov.6 ⊢ 𝐹 = { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ ( ( 𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆 ) ∧ 𝜑 ) }
7 df-ov ⊢ ( 𝐴 𝐹 𝐵 ) = ( 𝐹 ‘ ⟨ 𝐴 , 𝐵 ⟩ )
8 6 fveq1i ⊢ ( 𝐹 ‘ ⟨ 𝐴 , 𝐵 ⟩ ) = ( { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ ( ( 𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆 ) ∧ 𝜑 ) } ‘ ⟨ 𝐴 , 𝐵 ⟩ )
9 7 8 eqtri ⊢ ( 𝐴 𝐹 𝐵 ) = ( { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ ( ( 𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆 ) ∧ 𝜑 ) } ‘ ⟨ 𝐴 , 𝐵 ⟩ )
10 9 eqeq1i ⊢ ( ( 𝐴 𝐹 𝐵 ) = 𝐶 ↔ ( { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ ( ( 𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆 ) ∧ 𝜑 ) } ‘ ⟨ 𝐴 , 𝐵 ⟩ ) = 𝐶 )
11 5 fnoprab ⊢ { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ ( ( 𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆 ) ∧ 𝜑 ) } Fn { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆 ) }
12 eleq1 ⊢ ( 𝑥 = 𝐴 → ( 𝑥 ∈ 𝑅 ↔ 𝐴 ∈ 𝑅 ) )
13 12 anbi1d ⊢ ( 𝑥 = 𝐴 → ( ( 𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆 ) ↔ ( 𝐴 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆 ) ) )
14 eleq1 ⊢ ( 𝑦 = 𝐵 → ( 𝑦 ∈ 𝑆 ↔ 𝐵 ∈ 𝑆 ) )
15 14 anbi2d ⊢ ( 𝑦 = 𝐵 → ( ( 𝐴 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆 ) ↔ ( 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ) ) )
16 13 15 opelopabg ⊢ ( ( 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ) → ( ⟨ 𝐴 , 𝐵 ⟩ ∈ { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆 ) } ↔ ( 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ) ) )
17 16 ibir ⊢ ( ( 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ) → ⟨ 𝐴 , 𝐵 ⟩ ∈ { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆 ) } )
18 fnopfvb ⊢ ( ( { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ ( ( 𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆 ) ∧ 𝜑 ) } Fn { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆 ) } ∧ ⟨ 𝐴 , 𝐵 ⟩ ∈ { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆 ) } ) → ( ( { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ ( ( 𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆 ) ∧ 𝜑 ) } ‘ ⟨ 𝐴 , 𝐵 ⟩ ) = 𝐶 ↔ ⟨ ⟨ 𝐴 , 𝐵 ⟩ , 𝐶 ⟩ ∈ { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ ( ( 𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆 ) ∧ 𝜑 ) } ) )
19 11 17 18 sylancr ⊢ ( ( 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ) → ( ( { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ ( ( 𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆 ) ∧ 𝜑 ) } ‘ ⟨ 𝐴 , 𝐵 ⟩ ) = 𝐶 ↔ ⟨ ⟨ 𝐴 , 𝐵 ⟩ , 𝐶 ⟩ ∈ { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ ( ( 𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆 ) ∧ 𝜑 ) } ) )
20 13 2 anbi12d ⊢ ( 𝑥 = 𝐴 → ( ( ( 𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆 ) ∧ 𝜑 ) ↔ ( ( 𝐴 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆 ) ∧ 𝜓 ) ) )
21 15 3 anbi12d ⊢ ( 𝑦 = 𝐵 → ( ( ( 𝐴 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆 ) ∧ 𝜓 ) ↔ ( ( 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ) ∧ 𝜒 ) ) )
22 4 anbi2d ⊢ ( 𝑧 = 𝐶 → ( ( ( 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ) ∧ 𝜒 ) ↔ ( ( 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ) ∧ 𝜃 ) ) )
23 20 21 22 eloprabg ⊢ ( ( 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ V ) → ( ⟨ ⟨ 𝐴 , 𝐵 ⟩ , 𝐶 ⟩ ∈ { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ ( ( 𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆 ) ∧ 𝜑 ) } ↔ ( ( 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ) ∧ 𝜃 ) ) )
24 1 23 mp3an3 ⊢ ( ( 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ) → ( ⟨ ⟨ 𝐴 , 𝐵 ⟩ , 𝐶 ⟩ ∈ { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ ( ( 𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆 ) ∧ 𝜑 ) } ↔ ( ( 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ) ∧ 𝜃 ) ) )
25 19 24 bitrd ⊢ ( ( 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ) → ( ( { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ ( ( 𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆 ) ∧ 𝜑 ) } ‘ ⟨ 𝐴 , 𝐵 ⟩ ) = 𝐶 ↔ ( ( 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ) ∧ 𝜃 ) ) )
26 10 25 syl5bb ⊢ ( ( 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ) → ( ( 𝐴 𝐹 𝐵 ) = 𝐶 ↔ ( ( 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ) ∧ 𝜃 ) ) )
27 26 bianabs ⊢ ( ( 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ) → ( ( 𝐴 𝐹 𝐵 ) = 𝐶 ↔ 𝜃 ) )