Theorem isphtpc 23599

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 5031	. . 3 ⊢ (𝐹( ≃ph‘𝐽)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ ( ≃ph‘𝐽))
2		df-phtpc 23597	. . . 4 ⊢ ≃ph = (𝑗 ∈ Top ↦ {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ (II Cn 𝑗) ∧ (𝑓(PHtpy‘𝑗)𝑔) ≠ ∅)})
3	2	mptrcl 6754	. . 3 ⊢ (⟨𝐹, 𝐺⟩ ∈ ( ≃ph‘𝐽) → 𝐽 ∈ Top)
4	1, 3	sylbi 220	. 2 ⊢ (𝐹( ≃ph‘𝐽)𝐺 → 𝐽 ∈ Top)
5		cntop2 21846	. . 3 ⊢ (𝐹 ∈ (II Cn 𝐽) → 𝐽 ∈ Top)
6	5	3ad2ant1 1130	. 2 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ 𝐺 ∈ (II Cn 𝐽) ∧ (𝐹(PHtpy‘𝐽)𝐺) ≠ ∅) → 𝐽 ∈ Top)
7		oveq2 7143	. . . . . . . . 9 ⊢ (𝑗 = 𝐽 → (II Cn 𝑗) = (II Cn 𝐽))
8	7	sseq2d 3947	. . . . . . . 8 ⊢ (𝑗 = 𝐽 → ({𝑓, 𝑔} ⊆ (II Cn 𝑗) ↔ {𝑓, 𝑔} ⊆ (II Cn 𝐽)))
9		vex 3444	. . . . . . . . 9 ⊢ 𝑓 ∈ V
10		vex 3444	. . . . . . . . 9 ⊢ 𝑔 ∈ V
11	9, 10	prss 4713	. . . . . . . 8 ⊢ ((𝑓 ∈ (II Cn 𝐽) ∧ 𝑔 ∈ (II Cn 𝐽)) ↔ {𝑓, 𝑔} ⊆ (II Cn 𝐽))
12	8, 11	syl6bbr 292	. . . . . . 7 ⊢ (𝑗 = 𝐽 → ({𝑓, 𝑔} ⊆ (II Cn 𝑗) ↔ (𝑓 ∈ (II Cn 𝐽) ∧ 𝑔 ∈ (II Cn 𝐽))))
13		fveq2 6645	. . . . . . . . 9 ⊢ (𝑗 = 𝐽 → (PHtpy‘𝑗) = (PHtpy‘𝐽))
14	13	oveqd 7152	. . . . . . . 8 ⊢ (𝑗 = 𝐽 → (𝑓(PHtpy‘𝑗)𝑔) = (𝑓(PHtpy‘𝐽)𝑔))
15	14	neeq1d 3046	. . . . . . 7 ⊢ (𝑗 = 𝐽 → ((𝑓(PHtpy‘𝑗)𝑔) ≠ ∅ ↔ (𝑓(PHtpy‘𝐽)𝑔) ≠ ∅))
16	12, 15	anbi12d 633	. . . . . 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 5096	. . . . 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 7168	. . . . . . 7 ⊢ (II Cn 𝐽) ∈ V
19	18, 18	xpex 7456	. . . . . 6 ⊢ ((II Cn 𝐽) × (II Cn 𝐽)) ∈ V
20		opabssxp 5607	. . . . . 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 5190	. . . . 5 ⊢ {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (II Cn 𝐽) ∧ 𝑔 ∈ (II Cn 𝐽)) ∧ (𝑓(PHtpy‘𝐽)𝑔) ≠ ∅)} ∈ V
22	17, 2, 21	fvmpt 6745	. . . 4 ⊢ (𝐽 ∈ Top → ( ≃ph‘𝐽) = {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (II Cn 𝐽) ∧ 𝑔 ∈ (II Cn 𝐽)) ∧ (𝑓(PHtpy‘𝐽)𝑔) ≠ ∅)})
23	22	breqd 5041	. . 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 7144	. . . . . 6 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (𝑓(PHtpy‘𝐽)𝑔) = (𝐹(PHtpy‘𝐽)𝐺))
25	24	neeq1d 3046	. . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → ((𝑓(PHtpy‘𝐽)𝑔) ≠ ∅ ↔ (𝐹(PHtpy‘𝐽)𝐺) ≠ ∅))
26		eqid 2798	. . . . 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 5608	. . . 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 1086	. . . 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 281	. . 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	syl6bb 290	. 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 383	1 ⊢ (𝐹( ≃ph‘𝐽)𝐺 ↔ (𝐹 ∈ (II Cn 𝐽) ∧ 𝐺 ∈ (II Cn 𝐽) ∧ (𝐹(PHtpy‘𝐽)𝐺) ≠ ∅))

Syntax hints:   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   ≠ wne 2987   ⊆ wss 3881  ∅c0 4243  {cpr 4527  ⟨cop 4531   class class class wbr 5030  {copab 5092   × cxp 5517  ‘cfv 6324  (class class class)co 7135  Topctop 21498   Cn ccn 21829  IIcii 23480  PHtpycphtpy 23573   ≃_phcphtpc 23574

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  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

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-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-map 8391  df-top 21499  df-topon 21516  df-cn 21832  df-phtpc 23597

This theorem is referenced by:  phtpcer  23600  phtpc01  23601  reparpht  23603  phtpcco2  23604  pcohtpylem  23624  pcohtpy  23625  pcorevlem  23631  pi1blem  23644  txsconnlem  32600  txsconn  32601  cvxsconn  32603  cvmliftpht  32678