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Theorem htpycn 20557
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 20556 . . 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 457 . . 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 228 . 2  |-  ( ph  ->  ( h  e.  ( F ( J Htpy  K
) G )  ->  h  e.  ( ( J  tX  II )  Cn  K ) ) )
76ssrdv 3374 1  |-  ( ph  ->  ( F ( J Htpy 
K ) G ) 
C_  ( ( J 
tX  II )  Cn  K ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2727    C_ wss 3340   ` cfv 5430  (class class class)co 6103   0cc0 9294   1c1 9295  TopOnctopon 18511    Cn ccn 18840    tX ctx 19145   IIcii 20463   Htpy chtpy 20551
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-ral 2732  df-rex 2733  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-op 3896  df-uni 4104  df-iun 4185  df-br 4305  df-opab 4363  df-mpt 4364  df-id 4648  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-fv 5438  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-1st 6589  df-2nd 6590  df-map 7228  df-top 18515  df-topon 18518  df-cn 18843  df-htpy 20554
This theorem is referenced by:  htpycom  20560  htpyco1  20562  htpyco2  20563  htpycc  20564  phtpycn  20567
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