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Theorem htpycn 18471
Description: A homotopy is a continuous function. (Contributed by Mario Carneiro, 22-Feb-2015.)
Hypotheses
Ref Expression
ishtpy.1  |-  ( ph  ->  J  e.  (TopOn `  X ) )
ishtpy.3  |-  ( ph  ->  F  e.  ( J  Cn  K ) )
ishtpy.4  |-  ( ph  ->  G  e.  ( J  Cn  K ) )
Assertion
Ref Expression
htpycn  |-  ( ph  ->  ( F ( J Htpy 
K ) G ) 
C_  ( ( J 
tX  II )  Cn  K ) )

Proof of Theorem htpycn
Dummy variables  s  h are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ishtpy.1 . . . 4  |-  ( ph  ->  J  e.  (TopOn `  X ) )
2 ishtpy.3 . . . 4  |-  ( ph  ->  F  e.  ( J  Cn  K ) )
3 ishtpy.4 . . . 4  |-  ( ph  ->  G  e.  ( J  Cn  K ) )
41, 2, 3ishtpy 18470 . . 3  |-  ( ph  ->  ( h  e.  ( F ( J Htpy  K
) G )  <->  ( h  e.  ( ( J  tX  II )  Cn  K
)  /\  A. s  e.  X  ( (
s h 0 )  =  ( F `  s )  /\  (
s h 1 )  =  ( G `  s ) ) ) ) )
5 simpl 443 . . 3  |-  ( ( h  e.  ( ( J  tX  II )  Cn  K )  /\  A. s  e.  X  ( ( s h 0 )  =  ( F `
 s )  /\  ( s h 1 )  =  ( G `
 s ) ) )  ->  h  e.  ( ( J  tX  II )  Cn  K
) )
64, 5syl6bi 219 . 2  |-  ( ph  ->  ( h  e.  ( F ( J Htpy  K
) G )  ->  h  e.  ( ( J  tX  II )  Cn  K ) ) )
76ssrdv 3185 1  |-  ( ph  ->  ( F ( J Htpy 
K ) G ) 
C_  ( ( J 
tX  II )  Cn  K ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543    C_ wss 3152   ` cfv 5255  (class class class)co 5858   0cc0 8737   1c1 8738  TopOnctopon 16632    Cn ccn 16954    tX ctx 17255   IIcii 18379   Htpy chtpy 18465
This theorem is referenced by:  htpycom  18474  htpyco1  18476  htpyco2  18477  htpycc  18478  phtpycn  18481
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-map 6774  df-top 16636  df-topon 16639  df-cn 16957  df-htpy 18468
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