issconn - Mathbox for Mario Carneiro

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Theorem issconn 34841

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

**Assertion**
Ref	Expression
issconn	⊢ (𝐽 ∈ SConn ↔ (𝐽 ∈ PConn ∧ ∀𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = (𝑓‘1) → 𝑓( ≃_ph‘𝐽)((0[,]1) × {(𝑓‘0)}))))

Distinct variable group: 𝑓,𝐽

Proof of Theorem issconn Dummy variable 𝑗 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq2 7432	. . 3 ⊢ (𝑗 = 𝐽 → (II Cn 𝑗) = (II Cn 𝐽))
2		fveq2 6900	. . . . 5 ⊢ (𝑗 = 𝐽 → ( ≃_ph‘𝑗) = ( ≃_ph‘𝐽))
3	2	breqd 5161	. . . 4 ⊢ (𝑗 = 𝐽 → (𝑓( ≃_ph‘𝑗)((0[,]1) × {(𝑓‘0)}) ↔ 𝑓( ≃_ph‘𝐽)((0[,]1) × {(𝑓‘0)})))
4	3	imbi2d 339	. . 3 ⊢ (𝑗 = 𝐽 → (((𝑓‘0) = (𝑓‘1) → 𝑓( ≃_ph‘𝑗)((0[,]1) × {(𝑓‘0)})) ↔ ((𝑓‘0) = (𝑓‘1) → 𝑓( ≃_ph‘𝐽)((0[,]1) × {(𝑓‘0)}))))
5	1, 4	raleqbidv 3338	. 2 ⊢ (𝑗 = 𝐽 → (∀𝑓 ∈ (II Cn 𝑗)((𝑓‘0) = (𝑓‘1) → 𝑓( ≃_ph‘𝑗)((0[,]1) × {(𝑓‘0)})) ↔ ∀𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = (𝑓‘1) → 𝑓( ≃_ph‘𝐽)((0[,]1) × {(𝑓‘0)}))))
6		df-sconn 34837	. 2 ⊢ SConn = {𝑗 ∈ PConn ∣ ∀𝑓 ∈ (II Cn 𝑗)((𝑓‘0) = (𝑓‘1) → 𝑓( ≃_ph‘𝑗)((0[,]1) × {(𝑓‘0)}))}
7	5, 6	elrab2 3685	1 ⊢ (𝐽 ∈ SConn ↔ (𝐽 ∈ PConn ∧ ∀𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = (𝑓‘1) → 𝑓( ≃_ph‘𝐽)((0[,]1) × {(𝑓‘0)}))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∀wral 3057 {csn 4630 class class class wbr 5150 × cxp 5678 ‘cfv 6551 (class class class)co 7424 0cc0 11144 1c1 11145 [,]cicc 13365 Cn ccn 23146 IIcii 24813 ≃_phcphtpc 24913 PConncpconn 34834 SConncsconn 34835

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2698

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2705 df-cleq 2719 df-clel 2805 df-ral 3058 df-rab 3429 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4325 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4911 df-br 5151 df-iota 6503 df-fv 6559 df-ov 7427 df-sconn 34837

This theorem is referenced by: sconnpconn 34842 sconnpht 34844 sconnpi1 34854 txsconn 34856 cvxsconn 34858

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