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Theorem htpycn 21890
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 21889 . . 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 458 . . 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 231 . 2  |-  ( ph  ->  ( h  e.  ( F ( J Htpy  K
) G )  ->  h  e.  ( ( J  tX  II )  Cn  K ) ) )
76ssrdv 3467 1  |-  ( ph  ->  ( F ( J Htpy 
K ) G ) 
C_  ( ( J 
tX  II )  Cn  K ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1867   A.wral 2773    C_ wss 3433   ` cfv 5592  (class class class)co 6296   0cc0 9528   1c1 9529  TopOnctopon 19842    Cn ccn 20164    tX ctx 20499   IIcii 21796   Htpy chtpy 21884
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-iun 4295  df-br 4418  df-opab 4476  df-mpt 4477  df-id 4760  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-fv 5600  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6798  df-2nd 6799  df-map 7473  df-top 19845  df-topon 19847  df-cn 20167  df-htpy 21887
This theorem is referenced by:  htpycom  21893  htpyco1  21895  htpyco2  21896  htpycc  21897  phtpycn  21900
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