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Theorem phtpyi 24500
Description: Membership in the class of path homotopies between two continuous functions. (Contributed by Mario Carneiro, 23-Feb-2015.)
Hypotheses
Ref Expression
isphtpy.2 (𝜑𝐹 ∈ (II Cn 𝐽))
isphtpy.3 (𝜑𝐺 ∈ (II Cn 𝐽))
phtpyi.1 (𝜑𝐻 ∈ (𝐹(PHtpy‘𝐽)𝐺))
Assertion
Ref Expression
phtpyi ((𝜑𝐴 ∈ (0[,]1)) → ((0𝐻𝐴) = (𝐹‘0) ∧ (1𝐻𝐴) = (𝐹‘1)))

Proof of Theorem phtpyi
Dummy variable 𝑠 is distinct from all other variables.
StepHypRef Expression
1 phtpyi.1 . . . 4 (𝜑𝐻 ∈ (𝐹(PHtpy‘𝐽)𝐺))
2 isphtpy.2 . . . . 5 (𝜑𝐹 ∈ (II Cn 𝐽))
3 isphtpy.3 . . . . 5 (𝜑𝐺 ∈ (II Cn 𝐽))
42, 3isphtpy 24497 . . . 4 (𝜑 → (𝐻 ∈ (𝐹(PHtpy‘𝐽)𝐺) ↔ (𝐻 ∈ (𝐹(II Htpy 𝐽)𝐺) ∧ ∀𝑠 ∈ (0[,]1)((0𝐻𝑠) = (𝐹‘0) ∧ (1𝐻𝑠) = (𝐹‘1)))))
51, 4mpbid 231 . . 3 (𝜑 → (𝐻 ∈ (𝐹(II Htpy 𝐽)𝐺) ∧ ∀𝑠 ∈ (0[,]1)((0𝐻𝑠) = (𝐹‘0) ∧ (1𝐻𝑠) = (𝐹‘1))))
65simprd 497 . 2 (𝜑 → ∀𝑠 ∈ (0[,]1)((0𝐻𝑠) = (𝐹‘0) ∧ (1𝐻𝑠) = (𝐹‘1)))
7 oveq2 7417 . . . . 5 (𝑠 = 𝐴 → (0𝐻𝑠) = (0𝐻𝐴))
87eqeq1d 2735 . . . 4 (𝑠 = 𝐴 → ((0𝐻𝑠) = (𝐹‘0) ↔ (0𝐻𝐴) = (𝐹‘0)))
9 oveq2 7417 . . . . 5 (𝑠 = 𝐴 → (1𝐻𝑠) = (1𝐻𝐴))
109eqeq1d 2735 . . . 4 (𝑠 = 𝐴 → ((1𝐻𝑠) = (𝐹‘1) ↔ (1𝐻𝐴) = (𝐹‘1)))
118, 10anbi12d 632 . . 3 (𝑠 = 𝐴 → (((0𝐻𝑠) = (𝐹‘0) ∧ (1𝐻𝑠) = (𝐹‘1)) ↔ ((0𝐻𝐴) = (𝐹‘0) ∧ (1𝐻𝐴) = (𝐹‘1))))
1211rspccva 3612 . 2 ((∀𝑠 ∈ (0[,]1)((0𝐻𝑠) = (𝐹‘0) ∧ (1𝐻𝑠) = (𝐹‘1)) ∧ 𝐴 ∈ (0[,]1)) → ((0𝐻𝐴) = (𝐹‘0) ∧ (1𝐻𝐴) = (𝐹‘1)))
136, 12sylan 581 1 ((𝜑𝐴 ∈ (0[,]1)) → ((0𝐻𝐴) = (𝐹‘0) ∧ (1𝐻𝐴) = (𝐹‘1)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  wral 3062  cfv 6544  (class class class)co 7409  0cc0 11110  1c1 11111  [,]cicc 13327   Cn ccn 22728  IIcii 24391   Htpy chtpy 24483  PHtpycphtpy 24484
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-1st 7975  df-2nd 7976  df-map 8822  df-top 22396  df-topon 22413  df-cn 22731  df-phtpy 24487
This theorem is referenced by:  phtpy01  24501  phtpycom  24504  phtpyco2  24506  phtpycc  24507  pcohtpylem  24535  txsconnlem  34231  cvmliftphtlem  34308  cvmliftpht  34309
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