Theorem eqinfd

Description: Sufficient condition for an element to be equal to the infimum. (Contributed by AV, 3-Sep-2020)

	Ref	Expression
Hypotheses	infexd.1	⊢ ( 𝜑 → 𝑅 Or 𝐴 )
	eqinfd.2	⊢ ( 𝜑 → 𝐶 ∈ 𝐴 )
	eqinfd.3	⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ) → ¬ 𝑦 𝑅 𝐶 )
	eqinfd.4	⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ 𝐴 ∧ 𝐶 𝑅 𝑦 ) ) → ∃ 𝑧 ∈ 𝐵 𝑧 𝑅 𝑦 )
Assertion	eqinfd	⊢ ( 𝜑 → inf ( 𝐵 , 𝐴 , 𝑅 ) = 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		infexd.1	⊢ ( 𝜑 → 𝑅 Or 𝐴 )
2		eqinfd.2	⊢ ( 𝜑 → 𝐶 ∈ 𝐴 )
3		eqinfd.3	⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ) → ¬ 𝑦 𝑅 𝐶 )
4		eqinfd.4	⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ 𝐴 ∧ 𝐶 𝑅 𝑦 ) ) → ∃ 𝑧 ∈ 𝐵 𝑧 𝑅 𝑦 )
5	3	ralrimiva	⊢ ( 𝜑 → ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 𝑅 𝐶 )
6	4	expr	⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐴 ) → ( 𝐶 𝑅 𝑦 → ∃ 𝑧 ∈ 𝐵 𝑧 𝑅 𝑦 ) )
7	6	ralrimiva	⊢ ( 𝜑 → ∀ 𝑦 ∈ 𝐴 ( 𝐶 𝑅 𝑦 → ∃ 𝑧 ∈ 𝐵 𝑧 𝑅 𝑦 ) )
8	1	eqinf	⊢ ( 𝜑 → ( ( 𝐶 ∈ 𝐴 ∧ ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 𝑅 𝐶 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝐶 𝑅 𝑦 → ∃ 𝑧 ∈ 𝐵 𝑧 𝑅 𝑦 ) ) → inf ( 𝐵 , 𝐴 , 𝑅 ) = 𝐶 ) )
9	2 5 7 8	mp3and	⊢ ( 𝜑 → inf ( 𝐵 , 𝐴 , 𝑅 ) = 𝐶 )