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Mathbox for Mario Carneiro |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ispconn | Structured version Visualization version GIF version |
Description: The property of being a path-connected topological space. (Contributed by Mario Carneiro, 11-Feb-2015.) |
Ref | Expression |
---|---|
ispconn.1 | ⊢ 𝑋 = ∪ 𝐽 |
Ref | Expression |
---|---|
ispconn | ⊢ (𝐽 ∈ PConn ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∃𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | unieq 4876 | . . . 4 ⊢ (𝑗 = 𝐽 → ∪ 𝑗 = ∪ 𝐽) | |
2 | ispconn.1 | . . . 4 ⊢ 𝑋 = ∪ 𝐽 | |
3 | 1, 2 | eqtr4di 2794 | . . 3 ⊢ (𝑗 = 𝐽 → ∪ 𝑗 = 𝑋) |
4 | oveq2 7364 | . . . . 5 ⊢ (𝑗 = 𝐽 → (II Cn 𝑗) = (II Cn 𝐽)) | |
5 | 4 | rexeqdv 3314 | . . . 4 ⊢ (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 3319 | . . 3 ⊢ (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 3319 | . 2 ⊢ (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 33755 | . 2 ⊢ PConn = {𝑗 ∈ Top ∣ ∀𝑥 ∈ ∪ 𝑗∀𝑦 ∈ ∪ 𝑗∃𝑓 ∈ (II Cn 𝑗)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)} | |
9 | 7, 8 | elrab2 3648 | 1 ⊢ (𝐽 ∈ PConn ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∃𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∀wral 3064 ∃wrex 3073 ∪ cuni 4865 ‘cfv 6496 (class class class)co 7356 0cc0 11050 1c1 11051 Topctop 22240 Cn ccn 22573 IIcii 24236 PConncpconn 33753 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2707 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2714 df-cleq 2728 df-clel 2814 df-ral 3065 df-rex 3074 df-rab 3408 df-v 3447 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-br 5106 df-iota 6448 df-fv 6504 df-ov 7359 df-pconn 33755 |
This theorem is referenced by: pconncn 33758 pconntop 33759 cnpconn 33764 txpconn 33766 ptpconn 33767 indispconn 33768 connpconn 33769 cvxpconn 33776 |
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