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Theorem isphtpc 23163
 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 4874 . . 3 (𝐹( ≃ph𝐽)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ ( ≃ph𝐽))
2 df-phtpc 23161 . . . . 5 ph = (𝑗 ∈ Top ↦ {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ (II Cn 𝑗) ∧ (𝑓(PHtpy‘𝑗)𝑔) ≠ ∅)})
32dmmptss 5872 . . . 4 dom ≃ph ⊆ Top
4 elfvdm 6465 . . . 4 (⟨𝐹, 𝐺⟩ ∈ ( ≃ph𝐽) → 𝐽 ∈ dom ≃ph)
53, 4sseldi 3825 . . 3 (⟨𝐹, 𝐺⟩ ∈ ( ≃ph𝐽) → 𝐽 ∈ Top)
61, 5sylbi 209 . 2 (𝐹( ≃ph𝐽)𝐺𝐽 ∈ Top)
7 cntop2 21416 . . 3 (𝐹 ∈ (II Cn 𝐽) → 𝐽 ∈ Top)
873ad2ant1 1167 . 2 ((𝐹 ∈ (II Cn 𝐽) ∧ 𝐺 ∈ (II Cn 𝐽) ∧ (𝐹(PHtpy‘𝐽)𝐺) ≠ ∅) → 𝐽 ∈ Top)
9 oveq2 6913 . . . . . . . . 9 (𝑗 = 𝐽 → (II Cn 𝑗) = (II Cn 𝐽))
109sseq2d 3858 . . . . . . . 8 (𝑗 = 𝐽 → ({𝑓, 𝑔} ⊆ (II Cn 𝑗) ↔ {𝑓, 𝑔} ⊆ (II Cn 𝐽)))
11 vex 3417 . . . . . . . . 9 𝑓 ∈ V
12 vex 3417 . . . . . . . . 9 𝑔 ∈ V
1311, 12prss 4569 . . . . . . . 8 ((𝑓 ∈ (II Cn 𝐽) ∧ 𝑔 ∈ (II Cn 𝐽)) ↔ {𝑓, 𝑔} ⊆ (II Cn 𝐽))
1410, 13syl6bbr 281 . . . . . . 7 (𝑗 = 𝐽 → ({𝑓, 𝑔} ⊆ (II Cn 𝑗) ↔ (𝑓 ∈ (II Cn 𝐽) ∧ 𝑔 ∈ (II Cn 𝐽))))
15 fveq2 6433 . . . . . . . . 9 (𝑗 = 𝐽 → (PHtpy‘𝑗) = (PHtpy‘𝐽))
1615oveqd 6922 . . . . . . . 8 (𝑗 = 𝐽 → (𝑓(PHtpy‘𝑗)𝑔) = (𝑓(PHtpy‘𝐽)𝑔))
1716neeq1d 3058 . . . . . . 7 (𝑗 = 𝐽 → ((𝑓(PHtpy‘𝑗)𝑔) ≠ ∅ ↔ (𝑓(PHtpy‘𝐽)𝑔) ≠ ∅))
1814, 17anbi12d 624 . . . . . 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 4939 . . . . 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 6937 . . . . . . 7 (II Cn 𝐽) ∈ V
2120, 20xpex 7223 . . . . . 6 ((II Cn 𝐽) × (II Cn 𝐽)) ∈ V
22 opabssxp 5428 . . . . . 6 {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (II Cn 𝐽) ∧ 𝑔 ∈ (II Cn 𝐽)) ∧ (𝑓(PHtpy‘𝐽)𝑔) ≠ ∅)} ⊆ ((II Cn 𝐽) × (II Cn 𝐽))
2321, 22ssexi 5028 . . . . 5 {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (II Cn 𝐽) ∧ 𝑔 ∈ (II Cn 𝐽)) ∧ (𝑓(PHtpy‘𝐽)𝑔) ≠ ∅)} ∈ V
2419, 2, 23fvmpt 6529 . . . 4 (𝐽 ∈ Top → ( ≃ph𝐽) = {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (II Cn 𝐽) ∧ 𝑔 ∈ (II Cn 𝐽)) ∧ (𝑓(PHtpy‘𝐽)𝑔) ≠ ∅)})
2524breqd 4884 . . 3 (𝐽 ∈ Top → (𝐹( ≃ph𝐽)𝐺𝐹{⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (II Cn 𝐽) ∧ 𝑔 ∈ (II Cn 𝐽)) ∧ (𝑓(PHtpy‘𝐽)𝑔) ≠ ∅)}𝐺))
26 oveq12 6914 . . . . . 6 ((𝑓 = 𝐹𝑔 = 𝐺) → (𝑓(PHtpy‘𝐽)𝑔) = (𝐹(PHtpy‘𝐽)𝐺))
2726neeq1d 3058 . . . . 5 ((𝑓 = 𝐹𝑔 = 𝐺) → ((𝑓(PHtpy‘𝐽)𝑔) ≠ ∅ ↔ (𝐹(PHtpy‘𝐽)𝐺) ≠ ∅))
28 eqid 2825 . . . . 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, 28brab2a 5429 . . . 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 1113 . . . 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 270 . . 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 279 . 2 (𝐽 ∈ Top → (𝐹( ≃ph𝐽)𝐺 ↔ (𝐹 ∈ (II Cn 𝐽) ∧ 𝐺 ∈ (II Cn 𝐽) ∧ (𝐹(PHtpy‘𝐽)𝐺) ≠ ∅)))
336, 8, 32pm5.21nii 370 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 198   ∧ wa 386   ∧ w3a 1111   = wceq 1656   ∈ wcel 2164   ≠ wne 2999   ⊆ wss 3798  ∅c0 4144  {cpr 4399  ⟨cop 4403   class class class wbr 4873  {copab 4935   × cxp 5340  dom cdm 5342  ‘cfv 6123  (class class class)co 6905  Topctop 21068   Cn ccn 21399  IIcii 23048  PHtpycphtpy 23137   ≃phcphtpc 23138 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-fv 6131  df-ov 6908  df-oprab 6909  df-mpt2 6910  df-map 8124  df-top 21069  df-topon 21086  df-cn 21402  df-phtpc 23161 This theorem is referenced by:  phtpcer  23164  phtpc01  23165  reparpht  23167  phtpcco2  23168  pcohtpylem  23188  pcohtpy  23189  pcorevlem  23195  pi1blem  23208  txsconnlem  31757  txsconn  31758  cvxsconn  31760  cvmliftpht  31835
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