Metamath Proof Explorer


Theorem 2mpo0

Description: If the operation value of the operation value of two nested maps-to notation is not empty, all involved arguments belong to the corresponding base classes of the maps-to notations. (Contributed by AV, 21-May-2021)

Ref Expression
Hypotheses 2mpo0.o 𝑂 = ( 𝑥𝐴 , 𝑦𝐵𝐸 )
2mpo0.u ( ( 𝑋𝐴𝑌𝐵 ) → ( 𝑋 𝑂 𝑌 ) = ( 𝑠𝐶 , 𝑡𝐷𝐹 ) )
Assertion 2mpo0 ( ¬ ( ( 𝑋𝐴𝑌𝐵 ) ∧ ( 𝑆𝐶𝑇𝐷 ) ) → ( 𝑆 ( 𝑋 𝑂 𝑌 ) 𝑇 ) = ∅ )

Proof

Step Hyp Ref Expression
1 2mpo0.o 𝑂 = ( 𝑥𝐴 , 𝑦𝐵𝐸 )
2 2mpo0.u ( ( 𝑋𝐴𝑌𝐵 ) → ( 𝑋 𝑂 𝑌 ) = ( 𝑠𝐶 , 𝑡𝐷𝐹 ) )
3 ianor ( ¬ ( ( 𝑋𝐴𝑌𝐵 ) ∧ ( 𝑆𝐶𝑇𝐷 ) ) ↔ ( ¬ ( 𝑋𝐴𝑌𝐵 ) ∨ ¬ ( 𝑆𝐶𝑇𝐷 ) ) )
4 1 mpondm0 ( ¬ ( 𝑋𝐴𝑌𝐵 ) → ( 𝑋 𝑂 𝑌 ) = ∅ )
5 4 oveqd ( ¬ ( 𝑋𝐴𝑌𝐵 ) → ( 𝑆 ( 𝑋 𝑂 𝑌 ) 𝑇 ) = ( 𝑆𝑇 ) )
6 0ov ( 𝑆𝑇 ) = ∅
7 5 6 eqtrdi ( ¬ ( 𝑋𝐴𝑌𝐵 ) → ( 𝑆 ( 𝑋 𝑂 𝑌 ) 𝑇 ) = ∅ )
8 notnotb ( ( 𝑋𝐴𝑌𝐵 ) ↔ ¬ ¬ ( 𝑋𝐴𝑌𝐵 ) )
9 2 adantr ( ( ( 𝑋𝐴𝑌𝐵 ) ∧ ¬ ( 𝑆𝐶𝑇𝐷 ) ) → ( 𝑋 𝑂 𝑌 ) = ( 𝑠𝐶 , 𝑡𝐷𝐹 ) )
10 9 oveqd ( ( ( 𝑋𝐴𝑌𝐵 ) ∧ ¬ ( 𝑆𝐶𝑇𝐷 ) ) → ( 𝑆 ( 𝑋 𝑂 𝑌 ) 𝑇 ) = ( 𝑆 ( 𝑠𝐶 , 𝑡𝐷𝐹 ) 𝑇 ) )
11 eqid ( 𝑠𝐶 , 𝑡𝐷𝐹 ) = ( 𝑠𝐶 , 𝑡𝐷𝐹 )
12 11 mpondm0 ( ¬ ( 𝑆𝐶𝑇𝐷 ) → ( 𝑆 ( 𝑠𝐶 , 𝑡𝐷𝐹 ) 𝑇 ) = ∅ )
13 12 adantl ( ( ( 𝑋𝐴𝑌𝐵 ) ∧ ¬ ( 𝑆𝐶𝑇𝐷 ) ) → ( 𝑆 ( 𝑠𝐶 , 𝑡𝐷𝐹 ) 𝑇 ) = ∅ )
14 10 13 eqtrd ( ( ( 𝑋𝐴𝑌𝐵 ) ∧ ¬ ( 𝑆𝐶𝑇𝐷 ) ) → ( 𝑆 ( 𝑋 𝑂 𝑌 ) 𝑇 ) = ∅ )
15 8 14 sylanbr ( ( ¬ ¬ ( 𝑋𝐴𝑌𝐵 ) ∧ ¬ ( 𝑆𝐶𝑇𝐷 ) ) → ( 𝑆 ( 𝑋 𝑂 𝑌 ) 𝑇 ) = ∅ )
16 7 15 jaoi3 ( ( ¬ ( 𝑋𝐴𝑌𝐵 ) ∨ ¬ ( 𝑆𝐶𝑇𝐷 ) ) → ( 𝑆 ( 𝑋 𝑂 𝑌 ) 𝑇 ) = ∅ )
17 3 16 sylbi ( ¬ ( ( 𝑋𝐴𝑌𝐵 ) ∧ ( 𝑆𝐶𝑇𝐷 ) ) → ( 𝑆 ( 𝑋 𝑂 𝑌 ) 𝑇 ) = ∅ )