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Theorem isphtpc 22733
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( ≃ph𝐽)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.
StepHypRef Expression
1 df-br 4624 . . 3 (𝐹( ≃ph𝐽)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ ( ≃ph𝐽))
2 df-phtpc 22731 . . . . 5 ph = (𝑗 ∈ Top ↦ {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ (II Cn 𝑗) ∧ (𝑓(PHtpy‘𝑗)𝑔) ≠ ∅)})
32dmmptss 5600 . . . 4 dom ≃ph ⊆ Top
4 elfvdm 6187 . . . 4 (⟨𝐹, 𝐺⟩ ∈ ( ≃ph𝐽) → 𝐽 ∈ dom ≃ph)
53, 4sseldi 3586 . . 3 (⟨𝐹, 𝐺⟩ ∈ ( ≃ph𝐽) → 𝐽 ∈ Top)
61, 5sylbi 207 . 2 (𝐹( ≃ph𝐽)𝐺𝐽 ∈ Top)
7 cntop2 20985 . . 3 (𝐹 ∈ (II Cn 𝐽) → 𝐽 ∈ Top)
873ad2ant1 1080 . 2 ((𝐹 ∈ (II Cn 𝐽) ∧ 𝐺 ∈ (II Cn 𝐽) ∧ (𝐹(PHtpy‘𝐽)𝐺) ≠ ∅) → 𝐽 ∈ Top)
9 oveq2 6623 . . . . . . . . 9 (𝑗 = 𝐽 → (II Cn 𝑗) = (II Cn 𝐽))
109sseq2d 3618 . . . . . . . 8 (𝑗 = 𝐽 → ({𝑓, 𝑔} ⊆ (II Cn 𝑗) ↔ {𝑓, 𝑔} ⊆ (II Cn 𝐽)))
11 vex 3193 . . . . . . . . 9 𝑓 ∈ V
12 vex 3193 . . . . . . . . 9 𝑔 ∈ V
1311, 12prss 4326 . . . . . . . 8 ((𝑓 ∈ (II Cn 𝐽) ∧ 𝑔 ∈ (II Cn 𝐽)) ↔ {𝑓, 𝑔} ⊆ (II Cn 𝐽))
1410, 13syl6bbr 278 . . . . . . 7 (𝑗 = 𝐽 → ({𝑓, 𝑔} ⊆ (II Cn 𝑗) ↔ (𝑓 ∈ (II Cn 𝐽) ∧ 𝑔 ∈ (II Cn 𝐽))))
15 fveq2 6158 . . . . . . . . 9 (𝑗 = 𝐽 → (PHtpy‘𝑗) = (PHtpy‘𝐽))
1615oveqd 6632 . . . . . . . 8 (𝑗 = 𝐽 → (𝑓(PHtpy‘𝑗)𝑔) = (𝑓(PHtpy‘𝐽)𝑔))
1716neeq1d 2849 . . . . . . 7 (𝑗 = 𝐽 → ((𝑓(PHtpy‘𝑗)𝑔) ≠ ∅ ↔ (𝑓(PHtpy‘𝐽)𝑔) ≠ ∅))
1814, 17anbi12d 746 . . . . . 6 (𝑗 = 𝐽 → (({𝑓, 𝑔} ⊆ (II Cn 𝑗) ∧ (𝑓(PHtpy‘𝑗)𝑔) ≠ ∅) ↔ ((𝑓 ∈ (II Cn 𝐽) ∧ 𝑔 ∈ (II Cn 𝐽)) ∧ (𝑓(PHtpy‘𝐽)𝑔) ≠ ∅)))
1918opabbidv 4688 . . . . 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) ≠ ∅)})
20 ovex 6643 . . . . . . 7 (II Cn 𝐽) ∈ V
2120, 20xpex 6927 . . . . . 6 ((II Cn 𝐽) × (II Cn 𝐽)) ∈ V
22 opabssxp 5164 . . . . . 6 {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (II Cn 𝐽) ∧ 𝑔 ∈ (II Cn 𝐽)) ∧ (𝑓(PHtpy‘𝐽)𝑔) ≠ ∅)} ⊆ ((II Cn 𝐽) × (II Cn 𝐽))
2321, 22ssexi 4773 . . . . 5 {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (II Cn 𝐽) ∧ 𝑔 ∈ (II Cn 𝐽)) ∧ (𝑓(PHtpy‘𝐽)𝑔) ≠ ∅)} ∈ V
2419, 2, 23fvmpt 6249 . . . 4 (𝐽 ∈ Top → ( ≃ph𝐽) = {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (II Cn 𝐽) ∧ 𝑔 ∈ (II Cn 𝐽)) ∧ (𝑓(PHtpy‘𝐽)𝑔) ≠ ∅)})
2524breqd 4634 . . 3 (𝐽 ∈ Top → (𝐹( ≃ph𝐽)𝐺𝐹{⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (II Cn 𝐽) ∧ 𝑔 ∈ (II Cn 𝐽)) ∧ (𝑓(PHtpy‘𝐽)𝑔) ≠ ∅)}𝐺))
26 oveq12 6624 . . . . . 6 ((𝑓 = 𝐹𝑔 = 𝐺) → (𝑓(PHtpy‘𝐽)𝑔) = (𝐹(PHtpy‘𝐽)𝐺))
2726neeq1d 2849 . . . . 5 ((𝑓 = 𝐹𝑔 = 𝐺) → ((𝑓(PHtpy‘𝐽)𝑔) ≠ ∅ ↔ (𝐹(PHtpy‘𝐽)𝐺) ≠ ∅))
28 eqid 2621 . . . . 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) ≠ ∅)}
2927, 28brab2ga 5165 . . . 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) ≠ ∅))
30 df-3an 1038 . . . 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) ≠ ∅))
3129, 30bitr4i 267 . . 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) ≠ ∅))
3225, 31syl6bb 276 . 2 (𝐽 ∈ Top → (𝐹( ≃ph𝐽)𝐺 ↔ (𝐹 ∈ (II Cn 𝐽) ∧ 𝐺 ∈ (II Cn 𝐽) ∧ (𝐹(PHtpy‘𝐽)𝐺) ≠ ∅)))
336, 8, 32pm5.21nii 368 1 (𝐹( ≃ph𝐽)𝐺 ↔ (𝐹 ∈ (II Cn 𝐽) ∧ 𝐺 ∈ (II Cn 𝐽) ∧ (𝐹(PHtpy‘𝐽)𝐺) ≠ ∅))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wne 2790  wss 3560  c0 3897  {cpr 4157  cop 4161   class class class wbr 4623  {copab 4682   × cxp 5082  dom cdm 5084  cfv 5857  (class class class)co 6615  Topctop 20638   Cn ccn 20968  IIcii 22618  PHtpycphtpy 22707  phcphtpc 22708
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-fv 5865  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-map 7819  df-top 20639  df-topon 20656  df-cn 20971  df-phtpc 22731
This theorem is referenced by:  phtpcer  22734  phtpcerOLD  22735  phtpc01  22736  reparpht  22738  phtpcco2  22739  pcohtpylem  22759  pcohtpy  22760  pcorevlem  22766  pi1blem  22779  txsconnlem  30983  txsconn  30984  cvxsconn  30986  cvmliftpht  31061
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