Theorem pconncn 35406

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
pconncn	⊢ ((𝐽 ∈ PConn ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ∃𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝐴 ∧ (𝑓‘1) = 𝐵))

Distinct variable groups:   𝐴,𝑓   𝐵,𝑓   𝑓,𝐽

Allowed substitution hint:   𝑋(𝑓)

Proof of Theorem pconncn

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ispconn.1	. . . . 5 ⊢ 𝑋 = ∪ 𝐽
2	1	ispconn 35405	. . . 4 ⊢ (𝐽 ∈ PConn ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∃𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)))
3	2	simprbi 497	. . 3 ⊢ (𝐽 ∈ PConn → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∃𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦))
4		eqeq2 2748	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((𝑓‘0) = 𝑥 ↔ (𝑓‘0) = 𝐴))
5	4	anbi1d 632	. . . . 5 ⊢ (𝑥 = 𝐴 → (((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦) ↔ ((𝑓‘0) = 𝐴 ∧ (𝑓‘1) = 𝑦)))
6	5	rexbidv 3161	. . . 4 ⊢ (𝑥 = 𝐴 → (∃𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦) ↔ ∃𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝐴 ∧ (𝑓‘1) = 𝑦)))
7		eqeq2 2748	. . . . . 6 ⊢ (𝑦 = 𝐵 → ((𝑓‘1) = 𝑦 ↔ (𝑓‘1) = 𝐵))
8	7	anbi2d 631	. . . . 5 ⊢ (𝑦 = 𝐵 → (((𝑓‘0) = 𝐴 ∧ (𝑓‘1) = 𝑦) ↔ ((𝑓‘0) = 𝐴 ∧ (𝑓‘1) = 𝐵)))
9	8	rexbidv 3161	. . . 4 ⊢ (𝑦 = 𝐵 → (∃𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝐴 ∧ (𝑓‘1) = 𝑦) ↔ ∃𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝐴 ∧ (𝑓‘1) = 𝐵)))
10	6, 9	rspc2v 3575	. . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∃𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦) → ∃𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝐴 ∧ (𝑓‘1) = 𝐵)))
11	3, 10	syl5com 31	. 2 ⊢ (𝐽 ∈ PConn → ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ∃𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝐴 ∧ (𝑓‘1) = 𝐵)))
12	11	3impib 1117	1 ⊢ ((𝐽 ∈ PConn ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ∃𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝐴 ∧ (𝑓‘1) = 𝐵))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3051  ∃wrex 3061  ∪ cuni 4850  ‘cfv 6498  (class class class)co 7367  0cc0 11038  1c1 11039  Topctop 22858   Cn ccn 23189  IIcii 24842  PConncpconn 35401

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-iota 6454  df-fv 6506  df-ov 7370  df-pconn 35403

This theorem is referenced by:  cnpconn  35412  pconnconn  35413  txpconn  35414  ptpconn  35415  connpconn  35417  pconnpi1  35419  cvmlift3lem2  35502  cvmlift3lem7  35507