Theorem pi1bas3

Description: The base set of the fundamental group. (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 ) )
	pi1bas3.r	⊢ 𝑅 = ( ( ≃ph ‘ 𝐽 ) ∩ ( ∪ 𝐵 × ∪ 𝐵 ) )
Assertion	pi1bas3	⊢ ( 𝜑 → 𝐵 = ( ∪ 𝐵 / 𝑅 ) )

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		pi1bas3.r	⊢ 𝑅 = ( ( ≃ph ‘ 𝐽 ) ∩ ( ∪ 𝐵 × ∪ 𝐵 ) )
6	1 2 3 4	pi1bas2	⊢ ( 𝜑 → 𝐵 = ( ∪ 𝐵 / ( ≃ph ‘ 𝐽 ) ) )
7		eqid	⊢ ( 𝐽 Ω1 𝑌 ) = ( 𝐽 Ω1 𝑌 )
8		eqidd	⊢ ( 𝜑 → ( Base ‘ ( 𝐽 Ω1 𝑌 ) ) = ( Base ‘ ( 𝐽 Ω1 𝑌 ) ) )
9	1 2 3 7 4 8	pi1buni	⊢ ( 𝜑 → ∪ 𝐵 = ( Base ‘ ( 𝐽 Ω1 𝑌 ) ) )
10	1 2 3 7 4 9	pi1blem	⊢ ( 𝜑 → ( ( ( ≃ph ‘ 𝐽 ) “ ∪ 𝐵 ) ⊆ ∪ 𝐵 ∧ ∪ 𝐵 ⊆ ( II Cn 𝐽 ) ) )
11	10	simpld	⊢ ( 𝜑 → ( ( ≃ph ‘ 𝐽 ) “ ∪ 𝐵 ) ⊆ ∪ 𝐵 )
12		qsinxp	⊢ ( ( ( ≃ph ‘ 𝐽 ) “ ∪ 𝐵 ) ⊆ ∪ 𝐵 → ( ∪ 𝐵 / ( ≃ph ‘ 𝐽 ) ) = ( ∪ 𝐵 / ( ( ≃ph ‘ 𝐽 ) ∩ ( ∪ 𝐵 × ∪ 𝐵 ) ) ) )
13	11 12	syl	⊢ ( 𝜑 → ( ∪ 𝐵 / ( ≃ph ‘ 𝐽 ) ) = ( ∪ 𝐵 / ( ( ≃ph ‘ 𝐽 ) ∩ ( ∪ 𝐵 × ∪ 𝐵 ) ) ) )
14	6 13	eqtrd	⊢ ( 𝜑 → 𝐵 = ( ∪ 𝐵 / ( ( ≃ph ‘ 𝐽 ) ∩ ( ∪ 𝐵 × ∪ 𝐵 ) ) ) )
15		qseq2	⊢ ( 𝑅 = ( ( ≃ph ‘ 𝐽 ) ∩ ( ∪ 𝐵 × ∪ 𝐵 ) ) → ( ∪ 𝐵 / 𝑅 ) = ( ∪ 𝐵 / ( ( ≃ph ‘ 𝐽 ) ∩ ( ∪ 𝐵 × ∪ 𝐵 ) ) ) )
16	5 15	ax-mp	⊢ ( ∪ 𝐵 / 𝑅 ) = ( ∪ 𝐵 / ( ( ≃ph ‘ 𝐽 ) ∩ ( ∪ 𝐵 × ∪ 𝐵 ) ) )
17	14 16	eqtr4di	⊢ ( 𝜑 → 𝐵 = ( ∪ 𝐵 / 𝑅 ) )