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Theorem isphtpc 23120
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 4845 . . 3 (𝐹( ≃ph𝐽)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ ( ≃ph𝐽))
2 df-phtpc 23118 . . . . 5 ph = (𝑗 ∈ Top ↦ {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ (II Cn 𝑗) ∧ (𝑓(PHtpy‘𝑗)𝑔) ≠ ∅)})
32dmmptss 5851 . . . 4 dom ≃ph ⊆ Top
4 elfvdm 6444 . . . 4 (⟨𝐹, 𝐺⟩ ∈ ( ≃ph𝐽) → 𝐽 ∈ dom ≃ph)
53, 4sseldi 3797 . . 3 (⟨𝐹, 𝐺⟩ ∈ ( ≃ph𝐽) → 𝐽 ∈ Top)
61, 5sylbi 209 . 2 (𝐹( ≃ph𝐽)𝐺𝐽 ∈ Top)
7 cntop2 21373 . . 3 (𝐹 ∈ (II Cn 𝐽) → 𝐽 ∈ Top)
873ad2ant1 1164 . 2 ((𝐹 ∈ (II Cn 𝐽) ∧ 𝐺 ∈ (II Cn 𝐽) ∧ (𝐹(PHtpy‘𝐽)𝐺) ≠ ∅) → 𝐽 ∈ Top)
9 oveq2 6887 . . . . . . . . 9 (𝑗 = 𝐽 → (II Cn 𝑗) = (II Cn 𝐽))
109sseq2d 3830 . . . . . . . 8 (𝑗 = 𝐽 → ({𝑓, 𝑔} ⊆ (II Cn 𝑗) ↔ {𝑓, 𝑔} ⊆ (II Cn 𝐽)))
11 vex 3389 . . . . . . . . 9 𝑓 ∈ V
12 vex 3389 . . . . . . . . 9 𝑔 ∈ V
1311, 12prss 4540 . . . . . . . 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 6412 . . . . . . . . 9 (𝑗 = 𝐽 → (PHtpy‘𝑗) = (PHtpy‘𝐽))
1615oveqd 6896 . . . . . . . 8 (𝑗 = 𝐽 → (𝑓(PHtpy‘𝑗)𝑔) = (𝑓(PHtpy‘𝐽)𝑔))
1716neeq1d 3031 . . . . . . 7 (𝑗 = 𝐽 → ((𝑓(PHtpy‘𝑗)𝑔) ≠ ∅ ↔ (𝑓(PHtpy‘𝐽)𝑔) ≠ ∅))
1814, 17anbi12d 625 . . . . . 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 4910 . . . . 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 6911 . . . . . . 7 (II Cn 𝐽) ∈ V
2120, 20xpex 7197 . . . . . 6 ((II Cn 𝐽) × (II Cn 𝐽)) ∈ V
22 opabssxp 5399 . . . . . 6 {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (II Cn 𝐽) ∧ 𝑔 ∈ (II Cn 𝐽)) ∧ (𝑓(PHtpy‘𝐽)𝑔) ≠ ∅)} ⊆ ((II Cn 𝐽) × (II Cn 𝐽))
2321, 22ssexi 4999 . . . . 5 {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (II Cn 𝐽) ∧ 𝑔 ∈ (II Cn 𝐽)) ∧ (𝑓(PHtpy‘𝐽)𝑔) ≠ ∅)} ∈ V
2419, 2, 23fvmpt 6508 . . . 4 (𝐽 ∈ Top → ( ≃ph𝐽) = {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (II Cn 𝐽) ∧ 𝑔 ∈ (II Cn 𝐽)) ∧ (𝑓(PHtpy‘𝐽)𝑔) ≠ ∅)})
2524breqd 4855 . . 3 (𝐽 ∈ Top → (𝐹( ≃ph𝐽)𝐺𝐹{⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (II Cn 𝐽) ∧ 𝑔 ∈ (II Cn 𝐽)) ∧ (𝑓(PHtpy‘𝐽)𝑔) ≠ ∅)}𝐺))
26 oveq12 6888 . . . . . 6 ((𝑓 = 𝐹𝑔 = 𝐺) → (𝑓(PHtpy‘𝐽)𝑔) = (𝐹(PHtpy‘𝐽)𝐺))
2726neeq1d 3031 . . . . 5 ((𝑓 = 𝐹𝑔 = 𝐺) → ((𝑓(PHtpy‘𝐽)𝑔) ≠ ∅ ↔ (𝐹(PHtpy‘𝐽)𝐺) ≠ ∅))
28 eqid 2800 . . . . 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 5400 . . . 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 1110 . . . 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 385  w3a 1108   = wceq 1653  wcel 2157  wne 2972  wss 3770  c0 4116  {cpr 4371  cop 4375   class class class wbr 4844  {copab 4906   × cxp 5311  dom cdm 5313  cfv 6102  (class class class)co 6879  Topctop 21025   Cn ccn 21356  IIcii 23005  PHtpycphtpy 23094  phcphtpc 23095
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2378  ax-ext 2778  ax-sep 4976  ax-nul 4984  ax-pow 5036  ax-pr 5098  ax-un 7184
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2592  df-eu 2610  df-clab 2787  df-cleq 2793  df-clel 2796  df-nfc 2931  df-ne 2973  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3388  df-sbc 3635  df-dif 3773  df-un 3775  df-in 3777  df-ss 3784  df-nul 4117  df-if 4279  df-pw 4352  df-sn 4370  df-pr 4372  df-op 4376  df-uni 4630  df-br 4845  df-opab 4907  df-mpt 4924  df-id 5221  df-xp 5319  df-rel 5320  df-cnv 5321  df-co 5322  df-dm 5323  df-rn 5324  df-res 5325  df-ima 5326  df-iota 6065  df-fun 6104  df-fn 6105  df-f 6106  df-fv 6110  df-ov 6882  df-oprab 6883  df-mpt2 6884  df-map 8098  df-top 21026  df-topon 21043  df-cn 21359  df-phtpc 23118
This theorem is referenced by:  phtpcer  23121  phtpc01  23122  reparpht  23124  phtpcco2  23125  pcohtpylem  23145  pcohtpy  23146  pcorevlem  23152  pi1blem  23165  txsconnlem  31738  txsconn  31739  cvxsconn  31741  cvmliftpht  31816
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