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Theorem isphtpy 22981
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 21247 . . . . 5 (𝐹 ∈ (II Cn 𝐽) → 𝐽 ∈ Top)
3 oveq2 6821 . . . . . . 7 (𝑗 = 𝐽 → (II Cn 𝑗) = (II Cn 𝐽))
4 oveq2 6821 . . . . . . . . 9 (𝑗 = 𝐽 → (II Htpy 𝑗) = (II Htpy 𝐽))
54oveqd 6830 . . . . . . . 8 (𝑗 = 𝐽 → (𝑓(II Htpy 𝑗)𝑔) = (𝑓(II Htpy 𝐽)𝑔))
6 rabeq 3332 . . . . . . . 8 ((𝑓(II Htpy 𝑗)𝑔) = (𝑓(II Htpy 𝐽)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))})
75, 6syl 17 . . . . . . 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))})
83, 3, 7mpt2eq123dv 6882 . . . . . 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))}))
9 df-phtpy 22971 . . . . . 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))}))
10 ovex 6841 . . . . . . 7 (II Cn 𝐽) ∈ V
1110, 10mpt2ex 7415 . . . . . 6 (𝑓 ∈ (II Cn 𝐽), 𝑔 ∈ (II Cn 𝐽) ↦ { ∈ (𝑓(II Htpy 𝐽)𝑔) ∣ ∀𝑠 ∈ (0[,]1)((0𝑠) = (𝑓‘0) ∧ (1𝑠) = (𝑓‘1))}) ∈ V
128, 9, 11fvmpt 6444 . . . . 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))}))
131, 2, 123syl 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))}))
14 oveq12 6822 . . . . . 6 ((𝑓 = 𝐹𝑔 = 𝐺) → (𝑓(II Htpy 𝐽)𝑔) = (𝐹(II Htpy 𝐽)𝐺))
15 simpl 474 . . . . . . . . . 10 ((𝑓 = 𝐹𝑔 = 𝐺) → 𝑓 = 𝐹)
1615fveq1d 6354 . . . . . . . . 9 ((𝑓 = 𝐹𝑔 = 𝐺) → (𝑓‘0) = (𝐹‘0))
1716eqeq2d 2770 . . . . . . . 8 ((𝑓 = 𝐹𝑔 = 𝐺) → ((0𝑠) = (𝑓‘0) ↔ (0𝑠) = (𝐹‘0)))
1815fveq1d 6354 . . . . . . . . 9 ((𝑓 = 𝐹𝑔 = 𝐺) → (𝑓‘1) = (𝐹‘1))
1918eqeq2d 2770 . . . . . . . 8 ((𝑓 = 𝐹𝑔 = 𝐺) → ((1𝑠) = (𝑓‘1) ↔ (1𝑠) = (𝐹‘1)))
2017, 19anbi12d 749 . . . . . . 7 ((𝑓 = 𝐹𝑔 = 𝐺) → (((0𝑠) = (𝑓‘0) ∧ (1𝑠) = (𝑓‘1)) ↔ ((0𝑠) = (𝐹‘0) ∧ (1𝑠) = (𝐹‘1))))
2120ralbidv 3124 . . . . . 6 ((𝑓 = 𝐹𝑔 = 𝐺) → (∀𝑠 ∈ (0[,]1)((0𝑠) = (𝑓‘0) ∧ (1𝑠) = (𝑓‘1)) ↔ ∀𝑠 ∈ (0[,]1)((0𝑠) = (𝐹‘0) ∧ (1𝑠) = (𝐹‘1))))
2214, 21rabeqbidv 3335 . . . . 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))})
2322adantl 473 . . . 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))})
24 isphtpy.3 . . . 4 (𝜑𝐺 ∈ (II Cn 𝐽))
25 ovex 6841 . . . . . 6 (𝐹(II Htpy 𝐽)𝐺) ∈ V
2625rabex 4964 . . . . 5 { ∈ (𝐹(II Htpy 𝐽)𝐺) ∣ ∀𝑠 ∈ (0[,]1)((0𝑠) = (𝐹‘0) ∧ (1𝑠) = (𝐹‘1))} ∈ V
2726a1i 11 . . . 4 (𝜑 → { ∈ (𝐹(II Htpy 𝐽)𝐺) ∣ ∀𝑠 ∈ (0[,]1)((0𝑠) = (𝐹‘0) ∧ (1𝑠) = (𝐹‘1))} ∈ V)
2813, 23, 1, 24, 27ovmpt2d 6953 . . 3 (𝜑 → (𝐹(PHtpy‘𝐽)𝐺) = { ∈ (𝐹(II Htpy 𝐽)𝐺) ∣ ∀𝑠 ∈ (0[,]1)((0𝑠) = (𝐹‘0) ∧ (1𝑠) = (𝐹‘1))})
2928eleq2d 2825 . 2 (𝜑 → (𝐻 ∈ (𝐹(PHtpy‘𝐽)𝐺) ↔ 𝐻 ∈ { ∈ (𝐹(II Htpy 𝐽)𝐺) ∣ ∀𝑠 ∈ (0[,]1)((0𝑠) = (𝐹‘0) ∧ (1𝑠) = (𝐹‘1))}))
30 oveq 6819 . . . . . 6 ( = 𝐻 → (0𝑠) = (0𝐻𝑠))
3130eqeq1d 2762 . . . . 5 ( = 𝐻 → ((0𝑠) = (𝐹‘0) ↔ (0𝐻𝑠) = (𝐹‘0)))
32 oveq 6819 . . . . . 6 ( = 𝐻 → (1𝑠) = (1𝐻𝑠))
3332eqeq1d 2762 . . . . 5 ( = 𝐻 → ((1𝑠) = (𝐹‘1) ↔ (1𝐻𝑠) = (𝐹‘1)))
3431, 33anbi12d 749 . . . 4 ( = 𝐻 → (((0𝑠) = (𝐹‘0) ∧ (1𝑠) = (𝐹‘1)) ↔ ((0𝐻𝑠) = (𝐹‘0) ∧ (1𝐻𝑠) = (𝐹‘1))))
3534ralbidv 3124 . . 3 ( = 𝐻 → (∀𝑠 ∈ (0[,]1)((0𝑠) = (𝐹‘0) ∧ (1𝑠) = (𝐹‘1)) ↔ ∀𝑠 ∈ (0[,]1)((0𝐻𝑠) = (𝐹‘0) ∧ (1𝐻𝑠) = (𝐹‘1))))
3635elrab 3504 . 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))))
3729, 36syl6bb 276 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 196  wa 383   = wceq 1632  wcel 2139  wral 3050  {crab 3054  Vcvv 3340  cfv 6049  (class class class)co 6813  cmpt2 6815  0cc0 10128  1c1 10129  [,]cicc 12371  Topctop 20900   Cn ccn 21230  IIcii 22879   Htpy chtpy 22967  PHtpycphtpy 22968
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-1st 7333  df-2nd 7334  df-map 8025  df-top 20901  df-topon 20918  df-cn 21233  df-phtpy 22971
This theorem is referenced by:  phtpyhtpy  22982  phtpyi  22984  isphtpyd  22986
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