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Theorem isphtpy 22588
 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 20855 . . . . 5 (𝐹 ∈ (II Cn 𝐽) → 𝐽 ∈ Top)
3 oveq2 6557 . . . . . . 7 (𝑗 = 𝐽 → (II Cn 𝑗) = (II Cn 𝐽))
4 oveq2 6557 . . . . . . . . 9 (𝑗 = 𝐽 → (II Htpy 𝑗) = (II Htpy 𝐽))
54oveqd 6566 . . . . . . . 8 (𝑗 = 𝐽 → (𝑓(II Htpy 𝑗)𝑔) = (𝑓(II Htpy 𝐽)𝑔))
6 rabeq 3166 . . . . . . . 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 6615 . . . . . 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 22578 . . . . . 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 6577 . . . . . . 7 (II Cn 𝐽) ∈ V
1110, 10mpt2ex 7136 . . . . . 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 6191 . . . . 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 6558 . . . . . 6 ((𝑓 = 𝐹𝑔 = 𝐺) → (𝑓(II Htpy 𝐽)𝑔) = (𝐹(II Htpy 𝐽)𝐺))
15 simpl 472 . . . . . . . . . 10 ((𝑓 = 𝐹𝑔 = 𝐺) → 𝑓 = 𝐹)
1615fveq1d 6105 . . . . . . . . 9 ((𝑓 = 𝐹𝑔 = 𝐺) → (𝑓‘0) = (𝐹‘0))
1716eqeq2d 2620 . . . . . . . 8 ((𝑓 = 𝐹𝑔 = 𝐺) → ((0𝑠) = (𝑓‘0) ↔ (0𝑠) = (𝐹‘0)))
1815fveq1d 6105 . . . . . . . . 9 ((𝑓 = 𝐹𝑔 = 𝐺) → (𝑓‘1) = (𝐹‘1))
1918eqeq2d 2620 . . . . . . . 8 ((𝑓 = 𝐹𝑔 = 𝐺) → ((1𝑠) = (𝑓‘1) ↔ (1𝑠) = (𝐹‘1)))
2017, 19anbi12d 743 . . . . . . 7 ((𝑓 = 𝐹𝑔 = 𝐺) → (((0𝑠) = (𝑓‘0) ∧ (1𝑠) = (𝑓‘1)) ↔ ((0𝑠) = (𝐹‘0) ∧ (1𝑠) = (𝐹‘1))))
2120ralbidv 2969 . . . . . 6 ((𝑓 = 𝐹𝑔 = 𝐺) → (∀𝑠 ∈ (0[,]1)((0𝑠) = (𝑓‘0) ∧ (1𝑠) = (𝑓‘1)) ↔ ∀𝑠 ∈ (0[,]1)((0𝑠) = (𝐹‘0) ∧ (1𝑠) = (𝐹‘1))))
2214, 21rabeqbidv 3168 . . . . 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 481 . . . 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 6577 . . . . . 6 (𝐹(II Htpy 𝐽)𝐺) ∈ V
2625rabex 4740 . . . . 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 6686 . . 3 (𝜑 → (𝐹(PHtpy‘𝐽)𝐺) = { ∈ (𝐹(II Htpy 𝐽)𝐺) ∣ ∀𝑠 ∈ (0[,]1)((0𝑠) = (𝐹‘0) ∧ (1𝑠) = (𝐹‘1))})
2928eleq2d 2673 . 2 (𝜑 → (𝐻 ∈ (𝐹(PHtpy‘𝐽)𝐺) ↔ 𝐻 ∈ { ∈ (𝐹(II Htpy 𝐽)𝐺) ∣ ∀𝑠 ∈ (0[,]1)((0𝑠) = (𝐹‘0) ∧ (1𝑠) = (𝐹‘1))}))
30 oveq 6555 . . . . . 6 ( = 𝐻 → (0𝑠) = (0𝐻𝑠))
3130eqeq1d 2612 . . . . 5 ( = 𝐻 → ((0𝑠) = (𝐹‘0) ↔ (0𝐻𝑠) = (𝐹‘0)))
32 oveq 6555 . . . . . 6 ( = 𝐻 → (1𝑠) = (1𝐻𝑠))
3332eqeq1d 2612 . . . . 5 ( = 𝐻 → ((1𝑠) = (𝐹‘1) ↔ (1𝐻𝑠) = (𝐹‘1)))
3431, 33anbi12d 743 . . . 4 ( = 𝐻 → (((0𝑠) = (𝐹‘0) ∧ (1𝑠) = (𝐹‘1)) ↔ ((0𝐻𝑠) = (𝐹‘0) ∧ (1𝐻𝑠) = (𝐹‘1))))
3534ralbidv 2969 . . 3 ( = 𝐻 → (∀𝑠 ∈ (0[,]1)((0𝑠) = (𝐹‘0) ∧ (1𝑠) = (𝐹‘1)) ↔ ∀𝑠 ∈ (0[,]1)((0𝐻𝑠) = (𝐹‘0) ∧ (1𝐻𝑠) = (𝐹‘1))))
3635elrab 3331 . 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 275 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 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  {crab 2900  Vcvv 3173  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  0cc0 9815  1c1 9816  [,]cicc 12049  Topctop 20517   Cn ccn 20838  IIcii 22486   Htpy chtpy 22574  PHtpycphtpy 22575 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-map 7746  df-top 20521  df-topon 20523  df-cn 20841  df-phtpy 22578 This theorem is referenced by:  phtpyhtpy  22589  phtpyi  22591  isphtpyd  22593
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