Theorem mpoxopxnop0

Description: If the first argument of an operation given by a maps-to rule, where the first argument is a pair and the base set of the second argument is the first component of the first argument, is not an ordered pair, then the value of the operation is the empty set. (Contributed by Alexander van der Vekens, 10-Oct-2017)

	Ref	Expression
Hypothesis	mpoxopn0yelv.f	⊢ 𝐹 = ( 𝑥 ∈ V , 𝑦 ∈ ( 1st ‘ 𝑥 ) ↦ 𝐶 )
Assertion	mpoxopxnop0	⊢ ( ¬ 𝑉 ∈ ( V × V ) → ( 𝑉 𝐹 𝐾 ) = ∅ )

Proof

Step	Hyp	Ref	Expression
1		mpoxopn0yelv.f	⊢ 𝐹 = ( 𝑥 ∈ V , 𝑦 ∈ ( 1st ‘ 𝑥 ) ↦ 𝐶 )
2		neq0	⊢ ( ¬ ( 𝑉 𝐹 𝐾 ) = ∅ ↔ ∃ 𝑥 𝑥 ∈ ( 𝑉 𝐹 𝐾 ) )
3	1	dmmpossx	⊢ dom 𝐹 ⊆ ∪ 𝑥 ∈ V ( { 𝑥 } × ( 1st ‘ 𝑥 ) )
4		elfvdm	⊢ ( 𝑥 ∈ ( 𝐹 ‘ ⟨ 𝑉 , 𝐾 ⟩ ) → ⟨ 𝑉 , 𝐾 ⟩ ∈ dom 𝐹 )
5		df-ov	⊢ ( 𝑉 𝐹 𝐾 ) = ( 𝐹 ‘ ⟨ 𝑉 , 𝐾 ⟩ )
6	4 5	eleq2s	⊢ ( 𝑥 ∈ ( 𝑉 𝐹 𝐾 ) → ⟨ 𝑉 , 𝐾 ⟩ ∈ dom 𝐹 )
7	3 6	sselid	⊢ ( 𝑥 ∈ ( 𝑉 𝐹 𝐾 ) → ⟨ 𝑉 , 𝐾 ⟩ ∈ ∪ 𝑥 ∈ V ( { 𝑥 } × ( 1st ‘ 𝑥 ) ) )
8		fveq2	⊢ ( 𝑥 = 𝑉 → ( 1st ‘ 𝑥 ) = ( 1st ‘ 𝑉 ) )
9	8	opeliunxp2	⊢ ( ⟨ 𝑉 , 𝐾 ⟩ ∈ ∪ 𝑥 ∈ V ( { 𝑥 } × ( 1st ‘ 𝑥 ) ) ↔ ( 𝑉 ∈ V ∧ 𝐾 ∈ ( 1st ‘ 𝑉 ) ) )
10		eluni	⊢ ( 𝐾 ∈ ∪ dom { 𝑉 } ↔ ∃ 𝑛 ( 𝐾 ∈ 𝑛 ∧ 𝑛 ∈ dom { 𝑉 } ) )
11		ne0i	⊢ ( 𝑛 ∈ dom { 𝑉 } → dom { 𝑉 } ≠ ∅ )
12	11	ad2antlr	⊢ ( ( ( 𝐾 ∈ 𝑛 ∧ 𝑛 ∈ dom { 𝑉 } ) ∧ 𝑉 ∈ V ) → dom { 𝑉 } ≠ ∅ )
13		dmsnn0	⊢ ( 𝑉 ∈ ( V × V ) ↔ dom { 𝑉 } ≠ ∅ )
14	12 13	sylibr	⊢ ( ( ( 𝐾 ∈ 𝑛 ∧ 𝑛 ∈ dom { 𝑉 } ) ∧ 𝑉 ∈ V ) → 𝑉 ∈ ( V × V ) )
15	14	ex	⊢ ( ( 𝐾 ∈ 𝑛 ∧ 𝑛 ∈ dom { 𝑉 } ) → ( 𝑉 ∈ V → 𝑉 ∈ ( V × V ) ) )
16	15	exlimiv	⊢ ( ∃ 𝑛 ( 𝐾 ∈ 𝑛 ∧ 𝑛 ∈ dom { 𝑉 } ) → ( 𝑉 ∈ V → 𝑉 ∈ ( V × V ) ) )
17	10 16	sylbi	⊢ ( 𝐾 ∈ ∪ dom { 𝑉 } → ( 𝑉 ∈ V → 𝑉 ∈ ( V × V ) ) )
18		1stval	⊢ ( 1st ‘ 𝑉 ) = ∪ dom { 𝑉 }
19	17 18	eleq2s	⊢ ( 𝐾 ∈ ( 1st ‘ 𝑉 ) → ( 𝑉 ∈ V → 𝑉 ∈ ( V × V ) ) )
20	19	impcom	⊢ ( ( 𝑉 ∈ V ∧ 𝐾 ∈ ( 1st ‘ 𝑉 ) ) → 𝑉 ∈ ( V × V ) )
21	9 20	sylbi	⊢ ( ⟨ 𝑉 , 𝐾 ⟩ ∈ ∪ 𝑥 ∈ V ( { 𝑥 } × ( 1st ‘ 𝑥 ) ) → 𝑉 ∈ ( V × V ) )
22	7 21	syl	⊢ ( 𝑥 ∈ ( 𝑉 𝐹 𝐾 ) → 𝑉 ∈ ( V × V ) )
23	22	exlimiv	⊢ ( ∃ 𝑥 𝑥 ∈ ( 𝑉 𝐹 𝐾 ) → 𝑉 ∈ ( V × V ) )
24	2 23	sylbi	⊢ ( ¬ ( 𝑉 𝐹 𝐾 ) = ∅ → 𝑉 ∈ ( V × V ) )
25	24	con1i	⊢ ( ¬ 𝑉 ∈ ( V × V ) → ( 𝑉 𝐹 𝐾 ) = ∅ )