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Theorem ispconn 31533
Description: The property of being a path-connected topological space. (Contributed by Mario Carneiro, 11-Feb-2015.)
Hypothesis
Ref Expression
ispconn.1 𝑋 = 𝐽
Assertion
Ref Expression
ispconn (𝐽 ∈ PConn ↔ (𝐽 ∈ Top ∧ ∀𝑥𝑋𝑦𝑋𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)))
Distinct variable groups:   𝑥,𝑓,𝑦,𝐽   𝑥,𝑋,𝑦
Allowed substitution hint:   𝑋(𝑓)

Proof of Theorem ispconn
Dummy variable 𝑗 is distinct from all other variables.
StepHypRef Expression
1 unieq 4596 . . . 4 (𝑗 = 𝐽 𝑗 = 𝐽)
2 ispconn.1 . . . 4 𝑋 = 𝐽
31, 2syl6eqr 2812 . . 3 (𝑗 = 𝐽 𝑗 = 𝑋)
4 oveq2 6822 . . . . 5 (𝑗 = 𝐽 → (II Cn 𝑗) = (II Cn 𝐽))
54rexeqdv 3284 . . . 4 (𝑗 = 𝐽 → (∃𝑓 ∈ (II Cn 𝑗)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦) ↔ ∃𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)))
63, 5raleqbidv 3291 . . 3 (𝑗 = 𝐽 → (∀𝑦 𝑗𝑓 ∈ (II Cn 𝑗)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦) ↔ ∀𝑦𝑋𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)))
73, 6raleqbidv 3291 . 2 (𝑗 = 𝐽 → (∀𝑥 𝑗𝑦 𝑗𝑓 ∈ (II Cn 𝑗)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦) ↔ ∀𝑥𝑋𝑦𝑋𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)))
8 df-pconn 31531 . 2 PConn = {𝑗 ∈ Top ∣ ∀𝑥 𝑗𝑦 𝑗𝑓 ∈ (II Cn 𝑗)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)}
97, 8elrab2 3507 1 (𝐽 ∈ PConn ↔ (𝐽 ∈ Top ∧ ∀𝑥𝑋𝑦𝑋𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383   = wceq 1632  wcel 2139  wral 3050  wrex 3051   cuni 4588  cfv 6049  (class class class)co 6814  0cc0 10148  1c1 10149  Topctop 20920   Cn ccn 21250  IIcii 22899  PConncpconn 31529
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-iota 6012  df-fv 6057  df-ov 6817  df-pconn 31531
This theorem is referenced by:  pconncn  31534  pconntop  31535  cnpconn  31540  txpconn  31542  ptpconn  31543  indispconn  31544  connpconn  31545  cvxpconn  31552
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