Theorem isphtpc 25026

Description: The relation "is path homotopic to". (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 5-Sep-2015.)

Assertion

Ref	Expression
isphtpc	⊢ (𝐹( ≃ph‘𝐽)𝐺 ↔ (𝐹 ∈ (II Cn 𝐽) ∧ 𝐺 ∈ (II Cn 𝐽) ∧ (𝐹(PHtpy‘𝐽)𝐺) ≠ ∅))

Proof of Theorem isphtpc

Dummy variables 𝑓 𝑔 𝑗 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-br 5144	. . 3 ⊢ (𝐹( ≃ph‘𝐽)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ ( ≃ph‘𝐽))
2		df-phtpc 25024	. . . 4 ⊢ ≃ph = (𝑗 ∈ Top ↦ {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ (II Cn 𝑗) ∧ (𝑓(PHtpy‘𝑗)𝑔) ≠ ∅)})
3	2	mptrcl 7025	. . 3 ⊢ (⟨𝐹, 𝐺⟩ ∈ ( ≃ph‘𝐽) → 𝐽 ∈ Top)
4	1, 3	sylbi 217	. 2 ⊢ (𝐹( ≃ph‘𝐽)𝐺 → 𝐽 ∈ Top)
5		cntop2 23249	. . 3 ⊢ (𝐹 ∈ (II Cn 𝐽) → 𝐽 ∈ Top)
6	5	3ad2ant1 1134	. 2 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ 𝐺 ∈ (II Cn 𝐽) ∧ (𝐹(PHtpy‘𝐽)𝐺) ≠ ∅) → 𝐽 ∈ Top)
7		oveq2 7439	. . . . . . . . 9 ⊢ (𝑗 = 𝐽 → (II Cn 𝑗) = (II Cn 𝐽))
8	7	sseq2d 4016	. . . . . . . 8 ⊢ (𝑗 = 𝐽 → ({𝑓, 𝑔} ⊆ (II Cn 𝑗) ↔ {𝑓, 𝑔} ⊆ (II Cn 𝐽)))
9		vex 3484	. . . . . . . . 9 ⊢ 𝑓 ∈ V
10		vex 3484	. . . . . . . . 9 ⊢ 𝑔 ∈ V
11	9, 10	prss 4820	. . . . . . . 8 ⊢ ((𝑓 ∈ (II Cn 𝐽) ∧ 𝑔 ∈ (II Cn 𝐽)) ↔ {𝑓, 𝑔} ⊆ (II Cn 𝐽))
12	8, 11	bitr4di 289	. . . . . . 7 ⊢ (𝑗 = 𝐽 → ({𝑓, 𝑔} ⊆ (II Cn 𝑗) ↔ (𝑓 ∈ (II Cn 𝐽) ∧ 𝑔 ∈ (II Cn 𝐽))))
13		fveq2 6906	. . . . . . . . 9 ⊢ (𝑗 = 𝐽 → (PHtpy‘𝑗) = (PHtpy‘𝐽))
14	13	oveqd 7448	. . . . . . . 8 ⊢ (𝑗 = 𝐽 → (𝑓(PHtpy‘𝑗)𝑔) = (𝑓(PHtpy‘𝐽)𝑔))
15	14	neeq1d 3000	. . . . . . 7 ⊢ (𝑗 = 𝐽 → ((𝑓(PHtpy‘𝑗)𝑔) ≠ ∅ ↔ (𝑓(PHtpy‘𝐽)𝑔) ≠ ∅))
16	12, 15	anbi12d 632	. . . . . 6 ⊢ (𝑗 = 𝐽 → (({𝑓, 𝑔} ⊆ (II Cn 𝑗) ∧ (𝑓(PHtpy‘𝑗)𝑔) ≠ ∅) ↔ ((𝑓 ∈ (II Cn 𝐽) ∧ 𝑔 ∈ (II Cn 𝐽)) ∧ (𝑓(PHtpy‘𝐽)𝑔) ≠ ∅)))
17	16	opabbidv 5209	. . . . 5 ⊢ (𝑗 = 𝐽 → {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ (II Cn 𝑗) ∧ (𝑓(PHtpy‘𝑗)𝑔) ≠ ∅)} = {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (II Cn 𝐽) ∧ 𝑔 ∈ (II Cn 𝐽)) ∧ (𝑓(PHtpy‘𝐽)𝑔) ≠ ∅)})
18		ovex 7464	. . . . . . 7 ⊢ (II Cn 𝐽) ∈ V
19	18, 18	xpex 7773	. . . . . 6 ⊢ ((II Cn 𝐽) × (II Cn 𝐽)) ∈ V
20		opabssxp 5778	. . . . . 6 ⊢ {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (II Cn 𝐽) ∧ 𝑔 ∈ (II Cn 𝐽)) ∧ (𝑓(PHtpy‘𝐽)𝑔) ≠ ∅)} ⊆ ((II Cn 𝐽) × (II Cn 𝐽))
21	19, 20	ssexi 5322	. . . . 5 ⊢ {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (II Cn 𝐽) ∧ 𝑔 ∈ (II Cn 𝐽)) ∧ (𝑓(PHtpy‘𝐽)𝑔) ≠ ∅)} ∈ V
22	17, 2, 21	fvmpt 7016	. . . 4 ⊢ (𝐽 ∈ Top → ( ≃ph‘𝐽) = {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (II Cn 𝐽) ∧ 𝑔 ∈ (II Cn 𝐽)) ∧ (𝑓(PHtpy‘𝐽)𝑔) ≠ ∅)})
23	22	breqd 5154	. . 3 ⊢ (𝐽 ∈ Top → (𝐹( ≃ph‘𝐽)𝐺 ↔ 𝐹{⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (II Cn 𝐽) ∧ 𝑔 ∈ (II Cn 𝐽)) ∧ (𝑓(PHtpy‘𝐽)𝑔) ≠ ∅)}𝐺))
24		oveq12 7440	. . . . . 6 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (𝑓(PHtpy‘𝐽)𝑔) = (𝐹(PHtpy‘𝐽)𝐺))
25	24	neeq1d 3000	. . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → ((𝑓(PHtpy‘𝐽)𝑔) ≠ ∅ ↔ (𝐹(PHtpy‘𝐽)𝐺) ≠ ∅))
26		eqid 2737	. . . . 5 ⊢ {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (II Cn 𝐽) ∧ 𝑔 ∈ (II Cn 𝐽)) ∧ (𝑓(PHtpy‘𝐽)𝑔) ≠ ∅)} = {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (II Cn 𝐽) ∧ 𝑔 ∈ (II Cn 𝐽)) ∧ (𝑓(PHtpy‘𝐽)𝑔) ≠ ∅)}
27	25, 26	brab2a 5779	. . . 4 ⊢ (𝐹{⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (II Cn 𝐽) ∧ 𝑔 ∈ (II Cn 𝐽)) ∧ (𝑓(PHtpy‘𝐽)𝑔) ≠ ∅)}𝐺 ↔ ((𝐹 ∈ (II Cn 𝐽) ∧ 𝐺 ∈ (II Cn 𝐽)) ∧ (𝐹(PHtpy‘𝐽)𝐺) ≠ ∅))
28		df-3an 1089	. . . 4 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ 𝐺 ∈ (II Cn 𝐽) ∧ (𝐹(PHtpy‘𝐽)𝐺) ≠ ∅) ↔ ((𝐹 ∈ (II Cn 𝐽) ∧ 𝐺 ∈ (II Cn 𝐽)) ∧ (𝐹(PHtpy‘𝐽)𝐺) ≠ ∅))
29	27, 28	bitr4i 278	. . 3 ⊢ (𝐹{⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (II Cn 𝐽) ∧ 𝑔 ∈ (II Cn 𝐽)) ∧ (𝑓(PHtpy‘𝐽)𝑔) ≠ ∅)}𝐺 ↔ (𝐹 ∈ (II Cn 𝐽) ∧ 𝐺 ∈ (II Cn 𝐽) ∧ (𝐹(PHtpy‘𝐽)𝐺) ≠ ∅))
30	23, 29	bitrdi 287	. 2 ⊢ (𝐽 ∈ Top → (𝐹( ≃ph‘𝐽)𝐺 ↔ (𝐹 ∈ (II Cn 𝐽) ∧ 𝐺 ∈ (II Cn 𝐽) ∧ (𝐹(PHtpy‘𝐽)𝐺) ≠ ∅)))
31	4, 6, 30	pm5.21nii 378	1 ⊢ (𝐹( ≃ph‘𝐽)𝐺 ↔ (𝐹 ∈ (II Cn 𝐽) ∧ 𝐺 ∈ (II Cn 𝐽) ∧ (𝐹(PHtpy‘𝐽)𝐺) ≠ ∅))

Syntax hints:   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1540   ∈ wcel 2108   ≠ wne 2940   ⊆ wss 3951  ∅c0 4333  {cpr 4628  ⟨cop 4632   class class class wbr 5143  {copab 5205   × cxp 5683  ‘cfv 6561  (class class class)co 7431  Topctop 22899   Cn ccn 23232  IIcii 24901  PHtpycphtpy 25000   ≃_phcphtpc 25001

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-map 8868  df-top 22900  df-topon 22917  df-cn 23235  df-phtpc 25024

This theorem is referenced by:  phtpcer  25027  phtpc01  25028  reparpht  25031  phtpcco2  25032  pcohtpylem  25052  pcohtpy  25053  pcorevlem  25059  pi1blem  25072  txsconnlem  35245  txsconn  35246  cvxsconn  35248  cvmliftpht  35323