Theorem ispconn

Description: The property of being a path-connected topological space. (Contributed by Mario Carneiro, 11-Feb-2015)

	Ref	Expression
Hypothesis	ispconn.1	⊢ 𝑋 = ∪ 𝐽
Assertion	ispconn	⊢ ( 𝐽 ∈ PConn ↔ ( 𝐽 ∈ Top ∧ ∀ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ∃ 𝑓 ∈ ( II Cn 𝐽 ) ( ( 𝑓 ‘ 0 ) = 𝑥 ∧ ( 𝑓 ‘ 1 ) = 𝑦 ) ) )

Proof

Step	Hyp	Ref	Expression
1		ispconn.1	⊢ 𝑋 = ∪ 𝐽
2		unieq	⊢ ( 𝑗 = 𝐽 → ∪ 𝑗 = ∪ 𝐽 )
3	2 1	eqtr4di	⊢ ( 𝑗 = 𝐽 → ∪ 𝑗 = 𝑋 )
4		oveq2	⊢ ( 𝑗 = 𝐽 → ( II Cn 𝑗 ) = ( II Cn 𝐽 ) )
5	4	rexeqdv	⊢ ( 𝑗 = 𝐽 → ( ∃ 𝑓 ∈ ( II Cn 𝑗 ) ( ( 𝑓 ‘ 0 ) = 𝑥 ∧ ( 𝑓 ‘ 1 ) = 𝑦 ) ↔ ∃ 𝑓 ∈ ( II Cn 𝐽 ) ( ( 𝑓 ‘ 0 ) = 𝑥 ∧ ( 𝑓 ‘ 1 ) = 𝑦 ) ) )
6	3 5	raleqbidv	⊢ ( 𝑗 = 𝐽 → ( ∀ 𝑦 ∈ ∪ 𝑗 ∃ 𝑓 ∈ ( II Cn 𝑗 ) ( ( 𝑓 ‘ 0 ) = 𝑥 ∧ ( 𝑓 ‘ 1 ) = 𝑦 ) ↔ ∀ 𝑦 ∈ 𝑋 ∃ 𝑓 ∈ ( II Cn 𝐽 ) ( ( 𝑓 ‘ 0 ) = 𝑥 ∧ ( 𝑓 ‘ 1 ) = 𝑦 ) ) )
7	3 6	raleqbidv	⊢ ( 𝑗 = 𝐽 → ( ∀ 𝑥 ∈ ∪ 𝑗 ∀ 𝑦 ∈ ∪ 𝑗 ∃ 𝑓 ∈ ( II Cn 𝑗 ) ( ( 𝑓 ‘ 0 ) = 𝑥 ∧ ( 𝑓 ‘ 1 ) = 𝑦 ) ↔ ∀ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ∃ 𝑓 ∈ ( II Cn 𝐽 ) ( ( 𝑓 ‘ 0 ) = 𝑥 ∧ ( 𝑓 ‘ 1 ) = 𝑦 ) ) )
8		df-pconn	⊢ PConn = { 𝑗 ∈ Top ∣ ∀ 𝑥 ∈ ∪ 𝑗 ∀ 𝑦 ∈ ∪ 𝑗 ∃ 𝑓 ∈ ( II Cn 𝑗 ) ( ( 𝑓 ‘ 0 ) = 𝑥 ∧ ( 𝑓 ‘ 1 ) = 𝑦 ) }
9	7 8	elrab2	⊢ ( 𝐽 ∈ PConn ↔ ( 𝐽 ∈ Top ∧ ∀ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ∃ 𝑓 ∈ ( II Cn 𝐽 ) ( ( 𝑓 ‘ 0 ) = 𝑥 ∧ ( 𝑓 ‘ 1 ) = 𝑦 ) ) )