Theorem pi1eluni

Description: Elementhood in the base set of the loop space. (Contributed by Mario Carneiro, 10-Jul-2015)

	Ref	Expression
Hypotheses	pi1val.g	⊢ 𝐺 = ( 𝐽 π1 𝑌 )
	pi1val.1	⊢ ( 𝜑 → 𝐽 ∈ ( TopOn ‘ 𝑋 ) )
	pi1val.2	⊢ ( 𝜑 → 𝑌 ∈ 𝑋 )
	pi1bas2.b	⊢ ( 𝜑 → 𝐵 = ( Base ‘ 𝐺 ) )
Assertion	pi1eluni	⊢ ( 𝜑 → ( 𝐹 ∈ ∪ 𝐵 ↔ ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 0 ) = 𝑌 ∧ ( 𝐹 ‘ 1 ) = 𝑌 ) ) )

Proof

Step	Hyp	Ref	Expression
1		pi1val.g	⊢ 𝐺 = ( 𝐽 π1 𝑌 )
2		pi1val.1	⊢ ( 𝜑 → 𝐽 ∈ ( TopOn ‘ 𝑋 ) )
3		pi1val.2	⊢ ( 𝜑 → 𝑌 ∈ 𝑋 )
4		pi1bas2.b	⊢ ( 𝜑 → 𝐵 = ( Base ‘ 𝐺 ) )
5		eqid	⊢ ( 𝐽 Ω1 𝑌 ) = ( 𝐽 Ω1 𝑌 )
6		eqidd	⊢ ( 𝜑 → ( Base ‘ ( 𝐽 Ω1 𝑌 ) ) = ( Base ‘ ( 𝐽 Ω1 𝑌 ) ) )
7	1 2 3 5 4 6	pi1buni	⊢ ( 𝜑 → ∪ 𝐵 = ( Base ‘ ( 𝐽 Ω1 𝑌 ) ) )
8	5 2 3 7	om1elbas	⊢ ( 𝜑 → ( 𝐹 ∈ ∪ 𝐵 ↔ ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 0 ) = 𝑌 ∧ ( 𝐹 ‘ 1 ) = 𝑌 ) ) )