Theorem ispconn 32583

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 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∃𝑓 ∈ (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.

Step	Hyp	Ref	Expression
1		unieq 4811	. . . 4 ⊢ (𝑗 = 𝐽 → ∪ 𝑗 = ∪ 𝐽)
2		ispconn.1	. . . 4 ⊢ 𝑋 = ∪ 𝐽
3	1, 2	eqtr4di 2851	. . 3 ⊢ (𝑗 = 𝐽 → ∪ 𝑗 = 𝑋)
4		oveq2 7143	. . . . 5 ⊢ (𝑗 = 𝐽 → (II Cn 𝑗) = (II Cn 𝐽))
5	4	rexeqdv 3365	. . . 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 3354	. . 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 3354	. 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 32581	. 2 ⊢ PConn = {𝑗 ∈ Top ∣ ∀𝑥 ∈ ∪ 𝑗∀𝑦 ∈ ∪ 𝑗∃𝑓 ∈ (II Cn 𝑗)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)}
9	7, 8	elrab2 3631	1 ⊢ (𝐽 ∈ PConn ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∃𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)))

Syntax hints:   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3106  ∃wrex 3107  ∪ cuni 4800  ‘cfv 6324  (class class class)co 7135  0cc0 10526  1c1 10527  Topctop 21498   Cn ccn 21829  IIcii 23480  PConncpconn 32579

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-un 3886  df-in 3888  df-ss 3898  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-iota 6283  df-fv 6332  df-ov 7138  df-pconn 32581

This theorem is referenced by:  pconncn  32584  pconntop  32585  cnpconn  32590  txpconn  32592  ptpconn  32593  indispconn  32594  connpconn  32595  cvxpconn  32602