Theorem isphtpc 25110

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 5105	. . 3 ⊢ (𝐹( ≃ph‘𝐽)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ ( ≃ph‘𝐽))
2		df-phtpc 25108	. . . 4 ⊢ ≃ph = (𝑗 ∈ Top ↦ {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ (II Cn 𝑗) ∧ (𝑓(PHtpy‘𝑗)𝑔) ≠ ∅)})
3	2	mptrcl 6989	. . 3 ⊢ (⟨𝐹, 𝐺⟩ ∈ ( ≃ph‘𝐽) → 𝐽 ∈ Top)
4	1, 3	sylbi 220	. 2 ⊢ (𝐹( ≃ph‘𝐽)𝐺 → 𝐽 ∈ Top)
5		cntop2 23355	. . 3 ⊢ (𝐹 ∈ (II Cn 𝐽) → 𝐽 ∈ Top)
6	5	3ad2ant1 1149	. 2 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ 𝐺 ∈ (II Cn 𝐽) ∧ (𝐹(PHtpy‘𝐽)𝐺) ≠ ∅) → 𝐽 ∈ Top)
7		oveq2 7408	. . . . . . . . 9 ⊢ (𝑗 = 𝐽 → (II Cn 𝑗) = (II Cn 𝐽))
8	7	sseq2d 3971	. . . . . . . 8 ⊢ (𝑗 = 𝐽 → ({𝑓, 𝑔} ⊆ (II Cn 𝑗) ↔ {𝑓, 𝑔} ⊆ (II Cn 𝐽)))
9		vex 3461	. . . . . . . . 9 ⊢ 𝑓 ∈ V
10		vex 3461	. . . . . . . . 9 ⊢ 𝑔 ∈ V
11	9, 10	prss 4781	. . . . . . . 8 ⊢ ((𝑓 ∈ (II Cn 𝐽) ∧ 𝑔 ∈ (II Cn 𝐽)) ↔ {𝑓, 𝑔} ⊆ (II Cn 𝐽))
12	8, 11	bitr4di 292	. . . . . . 7 ⊢ (𝑗 = 𝐽 → ({𝑓, 𝑔} ⊆ (II Cn 𝑗) ↔ (𝑓 ∈ (II Cn 𝐽) ∧ 𝑔 ∈ (II Cn 𝐽))))
13		fveq2 6871	. . . . . . . . 9 ⊢ (𝑗 = 𝐽 → (PHtpy‘𝑗) = (PHtpy‘𝐽))
14	13	oveqd 7417	. . . . . . . 8 ⊢ (𝑗 = 𝐽 → (𝑓(PHtpy‘𝑗)𝑔) = (𝑓(PHtpy‘𝐽)𝑔))
15	14	neeq1d 3019	. . . . . . 7 ⊢ (𝑗 = 𝐽 → ((𝑓(PHtpy‘𝑗)𝑔) ≠ ∅ ↔ (𝑓(PHtpy‘𝐽)𝑔) ≠ ∅))
16	12, 15	anbi12d 643	. . . . . 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 5170	. . . . 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 7433	. . . . . . 7 ⊢ (II Cn 𝐽) ∈ V
19	18, 18	xpex 7740	. . . . . 6 ⊢ ((II Cn 𝐽) × (II Cn 𝐽)) ∈ V
20		opabssxp 5743	. . . . . 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 5282	. . . . 5 ⊢ {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (II Cn 𝐽) ∧ 𝑔 ∈ (II Cn 𝐽)) ∧ (𝑓(PHtpy‘𝐽)𝑔) ≠ ∅)} ∈ V
22	17, 2, 21	fvmpt 6979	. . . 4 ⊢ (𝐽 ∈ Top → ( ≃ph‘𝐽) = {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (II Cn 𝐽) ∧ 𝑔 ∈ (II Cn 𝐽)) ∧ (𝑓(PHtpy‘𝐽)𝑔) ≠ ∅)})
23	22	breqd 5115	. . 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 7409	. . . . . 6 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (𝑓(PHtpy‘𝐽)𝑔) = (𝐹(PHtpy‘𝐽)𝐺))
25	24	neeq1d 3019	. . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → ((𝑓(PHtpy‘𝐽)𝑔) ≠ ∅ ↔ (𝐹(PHtpy‘𝐽)𝐺) ≠ ∅))
26		eqid 2765	. . . . 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 5744	. . . 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 1103	. . . 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	bitrdi 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 381	1 ⊢ (𝐹( ≃ph‘𝐽)𝐺 ↔ (𝐹 ∈ (II Cn 𝐽) ∧ 𝐺 ∈ (II Cn 𝐽) ∧ (𝐹(PHtpy‘𝐽)𝐺) ≠ ∅))

Syntax hints:   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   = wceq 1563   ∈ wcel 2145   ≠ wne 2960   ⊆ wss 3907  ∅c0 4288  {cpr 4587  ⟨cop 4591   class class class wbr 5104  {copab 5166   × cxp 5649  ‘cfv 6525  (class class class)co 7400  Topctop 23007   Cn ccn 23338  IIcii 24991  PHtpycphtpy 25084   ≃_phcphtpc 25085

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5250  ax-nul 5260  ax-pow 5326  ax-pr 5394  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-sbc 3748  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5105  df-opab 5167  df-mpt 5186  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-fv 6533  df-ov 7403  df-oprab 7404  df-mpo 7405  df-map 8814  df-top 23008  df-topon 23025  df-cn 23341  df-phtpc 25108

This theorem is referenced by:  phtpcer  25111  phtpc01  25112  reparpht  25114  phtpcco2  25115  pcohtpylem  25135  pcohtpy  25136  pcorevlem  25142  pi1blem  25155  txsconnlem  35598  txsconn  35599  cvxsconn  35601  cvmliftpht  35676