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Theorem isphtpy 24344
Description: Membership in the class of path homotopies between two continuous functions. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 23-Feb-2015.)
Hypotheses
Ref Expression
isphtpy.2 (𝜑𝐹 ∈ (II Cn 𝐽))
isphtpy.3 (𝜑𝐺 ∈ (II Cn 𝐽))
Assertion
Ref Expression
isphtpy (𝜑 → (𝐻 ∈ (𝐹(PHtpy‘𝐽)𝐺) ↔ (𝐻 ∈ (𝐹(II Htpy 𝐽)𝐺) ∧ ∀𝑠 ∈ (0[,]1)((0𝐻𝑠) = (𝐹‘0) ∧ (1𝐻𝑠) = (𝐹‘1)))))
Distinct variable groups:   𝐹,𝑠   𝐺,𝑠   𝐻,𝑠   𝐽,𝑠   𝜑,𝑠

Proof of Theorem isphtpy
Dummy variables 𝑓 𝑔 𝑗 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isphtpy.2 . . . . 5 (𝜑𝐹 ∈ (II Cn 𝐽))
2 cntop2 22592 . . . . 5 (𝐹 ∈ (II Cn 𝐽) → 𝐽 ∈ Top)
3 oveq2 7365 . . . . . . 7 (𝑗 = 𝐽 → (II Cn 𝑗) = (II Cn 𝐽))
4 oveq2 7365 . . . . . . . . 9 (𝑗 = 𝐽 → (II Htpy 𝑗) = (II Htpy 𝐽))
54oveqd 7374 . . . . . . . 8 (𝑗 = 𝐽 → (𝑓(II Htpy 𝑗)𝑔) = (𝑓(II Htpy 𝐽)𝑔))
65rabeqdv 3422 . . . . . . 7 (𝑗 = 𝐽 → { ∈ (𝑓(II Htpy 𝑗)𝑔) ∣ ∀𝑠 ∈ (0[,]1)((0𝑠) = (𝑓‘0) ∧ (1𝑠) = (𝑓‘1))} = { ∈ (𝑓(II Htpy 𝐽)𝑔) ∣ ∀𝑠 ∈ (0[,]1)((0𝑠) = (𝑓‘0) ∧ (1𝑠) = (𝑓‘1))})
73, 3, 6mpoeq123dv 7432 . . . . . 6 (𝑗 = 𝐽 → (𝑓 ∈ (II Cn 𝑗), 𝑔 ∈ (II Cn 𝑗) ↦ { ∈ (𝑓(II Htpy 𝑗)𝑔) ∣ ∀𝑠 ∈ (0[,]1)((0𝑠) = (𝑓‘0) ∧ (1𝑠) = (𝑓‘1))}) = (𝑓 ∈ (II Cn 𝐽), 𝑔 ∈ (II Cn 𝐽) ↦ { ∈ (𝑓(II Htpy 𝐽)𝑔) ∣ ∀𝑠 ∈ (0[,]1)((0𝑠) = (𝑓‘0) ∧ (1𝑠) = (𝑓‘1))}))
8 df-phtpy 24334 . . . . . 6 PHtpy = (𝑗 ∈ Top ↦ (𝑓 ∈ (II Cn 𝑗), 𝑔 ∈ (II Cn 𝑗) ↦ { ∈ (𝑓(II Htpy 𝑗)𝑔) ∣ ∀𝑠 ∈ (0[,]1)((0𝑠) = (𝑓‘0) ∧ (1𝑠) = (𝑓‘1))}))
9 ovex 7390 . . . . . . 7 (II Cn 𝐽) ∈ V
109, 9mpoex 8012 . . . . . 6 (𝑓 ∈ (II Cn 𝐽), 𝑔 ∈ (II Cn 𝐽) ↦ { ∈ (𝑓(II Htpy 𝐽)𝑔) ∣ ∀𝑠 ∈ (0[,]1)((0𝑠) = (𝑓‘0) ∧ (1𝑠) = (𝑓‘1))}) ∈ V
117, 8, 10fvmpt 6948 . . . . 5 (𝐽 ∈ Top → (PHtpy‘𝐽) = (𝑓 ∈ (II Cn 𝐽), 𝑔 ∈ (II Cn 𝐽) ↦ { ∈ (𝑓(II Htpy 𝐽)𝑔) ∣ ∀𝑠 ∈ (0[,]1)((0𝑠) = (𝑓‘0) ∧ (1𝑠) = (𝑓‘1))}))
121, 2, 113syl 18 . . . 4 (𝜑 → (PHtpy‘𝐽) = (𝑓 ∈ (II Cn 𝐽), 𝑔 ∈ (II Cn 𝐽) ↦ { ∈ (𝑓(II Htpy 𝐽)𝑔) ∣ ∀𝑠 ∈ (0[,]1)((0𝑠) = (𝑓‘0) ∧ (1𝑠) = (𝑓‘1))}))
13 oveq12 7366 . . . . . 6 ((𝑓 = 𝐹𝑔 = 𝐺) → (𝑓(II Htpy 𝐽)𝑔) = (𝐹(II Htpy 𝐽)𝐺))
14 simpl 483 . . . . . . . . . 10 ((𝑓 = 𝐹𝑔 = 𝐺) → 𝑓 = 𝐹)
1514fveq1d 6844 . . . . . . . . 9 ((𝑓 = 𝐹𝑔 = 𝐺) → (𝑓‘0) = (𝐹‘0))
1615eqeq2d 2747 . . . . . . . 8 ((𝑓 = 𝐹𝑔 = 𝐺) → ((0𝑠) = (𝑓‘0) ↔ (0𝑠) = (𝐹‘0)))
1714fveq1d 6844 . . . . . . . . 9 ((𝑓 = 𝐹𝑔 = 𝐺) → (𝑓‘1) = (𝐹‘1))
1817eqeq2d 2747 . . . . . . . 8 ((𝑓 = 𝐹𝑔 = 𝐺) → ((1𝑠) = (𝑓‘1) ↔ (1𝑠) = (𝐹‘1)))
1916, 18anbi12d 631 . . . . . . 7 ((𝑓 = 𝐹𝑔 = 𝐺) → (((0𝑠) = (𝑓‘0) ∧ (1𝑠) = (𝑓‘1)) ↔ ((0𝑠) = (𝐹‘0) ∧ (1𝑠) = (𝐹‘1))))
2019ralbidv 3174 . . . . . 6 ((𝑓 = 𝐹𝑔 = 𝐺) → (∀𝑠 ∈ (0[,]1)((0𝑠) = (𝑓‘0) ∧ (1𝑠) = (𝑓‘1)) ↔ ∀𝑠 ∈ (0[,]1)((0𝑠) = (𝐹‘0) ∧ (1𝑠) = (𝐹‘1))))
2113, 20rabeqbidv 3424 . . . . 5 ((𝑓 = 𝐹𝑔 = 𝐺) → { ∈ (𝑓(II Htpy 𝐽)𝑔) ∣ ∀𝑠 ∈ (0[,]1)((0𝑠) = (𝑓‘0) ∧ (1𝑠) = (𝑓‘1))} = { ∈ (𝐹(II Htpy 𝐽)𝐺) ∣ ∀𝑠 ∈ (0[,]1)((0𝑠) = (𝐹‘0) ∧ (1𝑠) = (𝐹‘1))})
2221adantl 482 . . . 4 ((𝜑 ∧ (𝑓 = 𝐹𝑔 = 𝐺)) → { ∈ (𝑓(II Htpy 𝐽)𝑔) ∣ ∀𝑠 ∈ (0[,]1)((0𝑠) = (𝑓‘0) ∧ (1𝑠) = (𝑓‘1))} = { ∈ (𝐹(II Htpy 𝐽)𝐺) ∣ ∀𝑠 ∈ (0[,]1)((0𝑠) = (𝐹‘0) ∧ (1𝑠) = (𝐹‘1))})
23 isphtpy.3 . . . 4 (𝜑𝐺 ∈ (II Cn 𝐽))
24 ovex 7390 . . . . . 6 (𝐹(II Htpy 𝐽)𝐺) ∈ V
2524rabex 5289 . . . . 5 { ∈ (𝐹(II Htpy 𝐽)𝐺) ∣ ∀𝑠 ∈ (0[,]1)((0𝑠) = (𝐹‘0) ∧ (1𝑠) = (𝐹‘1))} ∈ V
2625a1i 11 . . . 4 (𝜑 → { ∈ (𝐹(II Htpy 𝐽)𝐺) ∣ ∀𝑠 ∈ (0[,]1)((0𝑠) = (𝐹‘0) ∧ (1𝑠) = (𝐹‘1))} ∈ V)
2712, 22, 1, 23, 26ovmpod 7507 . . 3 (𝜑 → (𝐹(PHtpy‘𝐽)𝐺) = { ∈ (𝐹(II Htpy 𝐽)𝐺) ∣ ∀𝑠 ∈ (0[,]1)((0𝑠) = (𝐹‘0) ∧ (1𝑠) = (𝐹‘1))})
2827eleq2d 2823 . 2 (𝜑 → (𝐻 ∈ (𝐹(PHtpy‘𝐽)𝐺) ↔ 𝐻 ∈ { ∈ (𝐹(II Htpy 𝐽)𝐺) ∣ ∀𝑠 ∈ (0[,]1)((0𝑠) = (𝐹‘0) ∧ (1𝑠) = (𝐹‘1))}))
29 oveq 7363 . . . . . 6 ( = 𝐻 → (0𝑠) = (0𝐻𝑠))
3029eqeq1d 2738 . . . . 5 ( = 𝐻 → ((0𝑠) = (𝐹‘0) ↔ (0𝐻𝑠) = (𝐹‘0)))
31 oveq 7363 . . . . . 6 ( = 𝐻 → (1𝑠) = (1𝐻𝑠))
3231eqeq1d 2738 . . . . 5 ( = 𝐻 → ((1𝑠) = (𝐹‘1) ↔ (1𝐻𝑠) = (𝐹‘1)))
3330, 32anbi12d 631 . . . 4 ( = 𝐻 → (((0𝑠) = (𝐹‘0) ∧ (1𝑠) = (𝐹‘1)) ↔ ((0𝐻𝑠) = (𝐹‘0) ∧ (1𝐻𝑠) = (𝐹‘1))))
3433ralbidv 3174 . . 3 ( = 𝐻 → (∀𝑠 ∈ (0[,]1)((0𝑠) = (𝐹‘0) ∧ (1𝑠) = (𝐹‘1)) ↔ ∀𝑠 ∈ (0[,]1)((0𝐻𝑠) = (𝐹‘0) ∧ (1𝐻𝑠) = (𝐹‘1))))
3534elrab 3645 . 2 (𝐻 ∈ { ∈ (𝐹(II Htpy 𝐽)𝐺) ∣ ∀𝑠 ∈ (0[,]1)((0𝑠) = (𝐹‘0) ∧ (1𝑠) = (𝐹‘1))} ↔ (𝐻 ∈ (𝐹(II Htpy 𝐽)𝐺) ∧ ∀𝑠 ∈ (0[,]1)((0𝐻𝑠) = (𝐹‘0) ∧ (1𝐻𝑠) = (𝐹‘1))))
3628, 35bitrdi 286 1 (𝜑 → (𝐻 ∈ (𝐹(PHtpy‘𝐽)𝐺) ↔ (𝐻 ∈ (𝐹(II Htpy 𝐽)𝐺) ∧ ∀𝑠 ∈ (0[,]1)((0𝐻𝑠) = (𝐹‘0) ∧ (1𝐻𝑠) = (𝐹‘1)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  wral 3064  {crab 3407  Vcvv 3445  cfv 6496  (class class class)co 7357  cmpo 7359  0cc0 11051  1c1 11052  [,]cicc 13267  Topctop 22242   Cn ccn 22575  IIcii 24238   Htpy chtpy 24330  PHtpycphtpy 24331
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-1st 7921  df-2nd 7922  df-map 8767  df-top 22243  df-topon 22260  df-cn 22578  df-phtpy 24334
This theorem is referenced by:  phtpyhtpy  24345  phtpyi  24347  isphtpyd  24349
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