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Theorem ishtpy 21598
Description: Membership in the class of homotopies between two continuous functions. (Contributed by Mario Carneiro, 22-Feb-2015.) (Revised by Mario Carneiro, 5-Sep-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
ishtpy  |-  ( 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 ) ) ) ) )
Distinct variable groups:    F, s    G, s    H, s    J, s    ph, s    X, s
Allowed substitution hint:    K( s)

Proof of Theorem ishtpy
Dummy variables  f 
g  h  j  k are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-htpy 21596 . . . . . 6  |- Htpy  =  ( j  e.  Top , 
k  e.  Top  |->  ( f  e.  ( j  Cn  k ) ,  g  e.  ( j  Cn  k )  |->  { h  e.  ( ( j  tX  II )  Cn  k )  | 
A. s  e.  U. j ( ( s h 0 )  =  ( f `  s
)  /\  ( s
h 1 )  =  ( g `  s
) ) } ) )
21a1i 11 . . . . 5  |-  ( ph  -> Htpy  =  ( j  e. 
Top ,  k  e.  Top  |->  ( f  e.  ( j  Cn  k
) ,  g  e.  ( j  Cn  k
)  |->  { h  e.  ( ( j  tX  II )  Cn  k
)  |  A. s  e.  U. j ( ( s h 0 )  =  ( f `  s )  /\  (
s h 1 )  =  ( g `  s ) ) } ) ) )
3 simprl 756 . . . . . . 7  |-  ( (
ph  /\  ( j  =  J  /\  k  =  K ) )  -> 
j  =  J )
4 simprr 757 . . . . . . 7  |-  ( (
ph  /\  ( j  =  J  /\  k  =  K ) )  -> 
k  =  K )
53, 4oveq12d 6314 . . . . . 6  |-  ( (
ph  /\  ( j  =  J  /\  k  =  K ) )  -> 
( j  Cn  k
)  =  ( J  Cn  K ) )
63oveq1d 6311 . . . . . . . 8  |-  ( (
ph  /\  ( j  =  J  /\  k  =  K ) )  -> 
( j  tX  II )  =  ( J  tX  II ) )
76, 4oveq12d 6314 . . . . . . 7  |-  ( (
ph  /\  ( j  =  J  /\  k  =  K ) )  -> 
( ( j  tX  II )  Cn  k
)  =  ( ( J  tX  II )  Cn  K ) )
83unieqd 4261 . . . . . . . . 9  |-  ( (
ph  /\  ( j  =  J  /\  k  =  K ) )  ->  U. j  =  U. J )
9 ishtpy.1 . . . . . . . . . . 11  |-  ( ph  ->  J  e.  (TopOn `  X ) )
10 toponuni 19555 . . . . . . . . . . 11  |-  ( J  e.  (TopOn `  X
)  ->  X  =  U. J )
119, 10syl 16 . . . . . . . . . 10  |-  ( ph  ->  X  =  U. J
)
1211adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  ( j  =  J  /\  k  =  K ) )  ->  X  =  U. J )
138, 12eqtr4d 2501 . . . . . . . 8  |-  ( (
ph  /\  ( j  =  J  /\  k  =  K ) )  ->  U. j  =  X
)
1413raleqdv 3060 . . . . . . 7  |-  ( (
ph  /\  ( j  =  J  /\  k  =  K ) )  -> 
( A. s  e. 
U. j ( ( s h 0 )  =  ( f `  s )  /\  (
s h 1 )  =  ( g `  s ) )  <->  A. s  e.  X  ( (
s h 0 )  =  ( f `  s )  /\  (
s h 1 )  =  ( g `  s ) ) ) )
157, 14rabeqbidv 3104 . . . . . 6  |-  ( (
ph  /\  ( j  =  J  /\  k  =  K ) )  ->  { h  e.  (
( j  tX  II )  Cn  k )  | 
A. s  e.  U. j ( ( s h 0 )  =  ( f `  s
)  /\  ( s
h 1 )  =  ( g `  s
) ) }  =  { h  e.  (
( J  tX  II )  Cn  K )  | 
A. s  e.  X  ( ( s h 0 )  =  ( f `  s )  /\  ( s h 1 )  =  ( g `  s ) ) } )
165, 5, 15mpt2eq123dv 6358 . . . . 5  |-  ( (
ph  /\  ( j  =  J  /\  k  =  K ) )  -> 
( f  e.  ( j  Cn  k ) ,  g  e.  ( j  Cn  k ) 
|->  { h  e.  ( ( j  tX  II )  Cn  k )  | 
A. s  e.  U. j ( ( s h 0 )  =  ( f `  s
)  /\  ( s
h 1 )  =  ( g `  s
) ) } )  =  ( f  e.  ( J  Cn  K
) ,  g  e.  ( J  Cn  K
)  |->  { h  e.  ( ( J  tX  II )  Cn  K
)  |  A. s  e.  X  ( (
s h 0 )  =  ( f `  s )  /\  (
s h 1 )  =  ( g `  s ) ) } ) )
17 topontop 19554 . . . . . 6  |-  ( J  e.  (TopOn `  X
)  ->  J  e.  Top )
189, 17syl 16 . . . . 5  |-  ( ph  ->  J  e.  Top )
19 ishtpy.3 . . . . . 6  |-  ( ph  ->  F  e.  ( J  Cn  K ) )
20 cntop2 19869 . . . . . 6  |-  ( F  e.  ( J  Cn  K )  ->  K  e.  Top )
2119, 20syl 16 . . . . 5  |-  ( ph  ->  K  e.  Top )
22 ssrab2 3581 . . . . . . . . . 10  |-  { h  e.  ( ( J  tX  II )  Cn  K
)  |  A. s  e.  X  ( (
s h 0 )  =  ( f `  s )  /\  (
s h 1 )  =  ( g `  s ) ) } 
C_  ( ( J 
tX  II )  Cn  K )
23 ovex 6324 . . . . . . . . . . 11  |-  ( ( J  tX  II )  Cn  K )  e. 
_V
2423elpw2 4620 . . . . . . . . . 10  |-  ( { h  e.  ( ( J  tX  II )  Cn  K )  | 
A. s  e.  X  ( ( s h 0 )  =  ( f `  s )  /\  ( s h 1 )  =  ( g `  s ) ) }  e.  ~P ( ( J  tX  II )  Cn  K
)  <->  { h  e.  ( ( J  tX  II )  Cn  K )  | 
A. s  e.  X  ( ( s h 0 )  =  ( f `  s )  /\  ( s h 1 )  =  ( g `  s ) ) }  C_  (
( J  tX  II )  Cn  K ) )
2522, 24mpbir 209 . . . . . . . . 9  |-  { h  e.  ( ( J  tX  II )  Cn  K
)  |  A. s  e.  X  ( (
s h 0 )  =  ( f `  s )  /\  (
s h 1 )  =  ( g `  s ) ) }  e.  ~P ( ( J  tX  II )  Cn  K )
2625rgen2w 2819 . . . . . . . 8  |-  A. f  e.  ( J  Cn  K
) A. g  e.  ( J  Cn  K
) { h  e.  ( ( J  tX  II )  Cn  K
)  |  A. s  e.  X  ( (
s h 0 )  =  ( f `  s )  /\  (
s h 1 )  =  ( g `  s ) ) }  e.  ~P ( ( J  tX  II )  Cn  K )
27 eqid 2457 . . . . . . . . 9  |-  ( f  e.  ( J  Cn  K ) ,  g  e.  ( J  Cn  K )  |->  { h  e.  ( ( J  tX  II )  Cn  K
)  |  A. s  e.  X  ( (
s h 0 )  =  ( f `  s )  /\  (
s h 1 )  =  ( g `  s ) ) } )  =  ( f  e.  ( J  Cn  K ) ,  g  e.  ( J  Cn  K )  |->  { h  e.  ( ( J  tX  II )  Cn  K
)  |  A. s  e.  X  ( (
s h 0 )  =  ( f `  s )  /\  (
s h 1 )  =  ( g `  s ) ) } )
2827fmpt2 6866 . . . . . . . 8  |-  ( A. f  e.  ( J  Cn  K ) A. g  e.  ( J  Cn  K
) { h  e.  ( ( J  tX  II )  Cn  K
)  |  A. s  e.  X  ( (
s h 0 )  =  ( f `  s )  /\  (
s h 1 )  =  ( g `  s ) ) }  e.  ~P ( ( J  tX  II )  Cn  K )  <->  ( f  e.  ( J  Cn  K
) ,  g  e.  ( J  Cn  K
)  |->  { h  e.  ( ( J  tX  II )  Cn  K
)  |  A. s  e.  X  ( (
s h 0 )  =  ( f `  s )  /\  (
s h 1 )  =  ( g `  s ) ) } ) : ( ( J  Cn  K )  X.  ( J  Cn  K ) ) --> ~P ( ( J  tX  II )  Cn  K
) )
2926, 28mpbi 208 . . . . . . 7  |-  ( f  e.  ( J  Cn  K ) ,  g  e.  ( J  Cn  K )  |->  { h  e.  ( ( J  tX  II )  Cn  K
)  |  A. s  e.  X  ( (
s h 0 )  =  ( f `  s )  /\  (
s h 1 )  =  ( g `  s ) ) } ) : ( ( J  Cn  K )  X.  ( J  Cn  K ) ) --> ~P ( ( J  tX  II )  Cn  K
)
30 ovex 6324 . . . . . . . 8  |-  ( J  Cn  K )  e. 
_V
3130, 30xpex 6603 . . . . . . 7  |-  ( ( J  Cn  K )  X.  ( J  Cn  K ) )  e. 
_V
3223pwex 4639 . . . . . . 7  |-  ~P (
( J  tX  II )  Cn  K )  e. 
_V
33 fex2 6754 . . . . . . 7  |-  ( ( ( f  e.  ( J  Cn  K ) ,  g  e.  ( J  Cn  K ) 
|->  { h  e.  ( ( J  tX  II )  Cn  K )  | 
A. s  e.  X  ( ( s h 0 )  =  ( f `  s )  /\  ( s h 1 )  =  ( g `  s ) ) } ) : ( ( J  Cn  K )  X.  ( J  Cn  K ) ) --> ~P ( ( J 
tX  II )  Cn  K )  /\  (
( J  Cn  K
)  X.  ( J  Cn  K ) )  e.  _V  /\  ~P ( ( J  tX  II )  Cn  K
)  e.  _V )  ->  ( f  e.  ( J  Cn  K ) ,  g  e.  ( J  Cn  K ) 
|->  { h  e.  ( ( J  tX  II )  Cn  K )  | 
A. s  e.  X  ( ( s h 0 )  =  ( f `  s )  /\  ( s h 1 )  =  ( g `  s ) ) } )  e. 
_V )
3429, 31, 32, 33mp3an 1324 . . . . . 6  |-  ( f  e.  ( J  Cn  K ) ,  g  e.  ( J  Cn  K )  |->  { h  e.  ( ( J  tX  II )  Cn  K
)  |  A. s  e.  X  ( (
s h 0 )  =  ( f `  s )  /\  (
s h 1 )  =  ( g `  s ) ) } )  e.  _V
3534a1i 11 . . . . 5  |-  ( ph  ->  ( f  e.  ( J  Cn  K ) ,  g  e.  ( J  Cn  K ) 
|->  { h  e.  ( ( J  tX  II )  Cn  K )  | 
A. s  e.  X  ( ( s h 0 )  =  ( f `  s )  /\  ( s h 1 )  =  ( g `  s ) ) } )  e. 
_V )
362, 16, 18, 21, 35ovmpt2d 6429 . . . 4  |-  ( ph  ->  ( J Htpy  K )  =  ( f  e.  ( J  Cn  K
) ,  g  e.  ( J  Cn  K
)  |->  { h  e.  ( ( J  tX  II )  Cn  K
)  |  A. s  e.  X  ( (
s h 0 )  =  ( f `  s )  /\  (
s h 1 )  =  ( g `  s ) ) } ) )
37 fveq1 5871 . . . . . . . . 9  |-  ( f  =  F  ->  (
f `  s )  =  ( F `  s ) )
3837eqeq2d 2471 . . . . . . . 8  |-  ( f  =  F  ->  (
( s h 0 )  =  ( f `
 s )  <->  ( s
h 0 )  =  ( F `  s
) ) )
39 fveq1 5871 . . . . . . . . 9  |-  ( g  =  G  ->  (
g `  s )  =  ( G `  s ) )
4039eqeq2d 2471 . . . . . . . 8  |-  ( g  =  G  ->  (
( s h 1 )  =  ( g `
 s )  <->  ( s
h 1 )  =  ( G `  s
) ) )
4138, 40bi2anan9 873 . . . . . . 7  |-  ( ( f  =  F  /\  g  =  G )  ->  ( ( ( s h 0 )  =  ( f `  s
)  /\  ( s
h 1 )  =  ( g `  s
) )  <->  ( (
s h 0 )  =  ( F `  s )  /\  (
s h 1 )  =  ( G `  s ) ) ) )
4241adantl 466 . . . . . 6  |-  ( (
ph  /\  ( f  =  F  /\  g  =  G ) )  -> 
( ( ( s h 0 )  =  ( f `  s
)  /\  ( s
h 1 )  =  ( g `  s
) )  <->  ( (
s h 0 )  =  ( F `  s )  /\  (
s h 1 )  =  ( G `  s ) ) ) )
4342ralbidv 2896 . . . . 5  |-  ( (
ph  /\  ( f  =  F  /\  g  =  G ) )  -> 
( A. s  e.  X  ( ( s h 0 )  =  ( f `  s
)  /\  ( s
h 1 )  =  ( g `  s
) )  <->  A. s  e.  X  ( (
s h 0 )  =  ( F `  s )  /\  (
s h 1 )  =  ( G `  s ) ) ) )
4443rabbidv 3101 . . . 4  |-  ( (
ph  /\  ( f  =  F  /\  g  =  G ) )  ->  { 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 )  | 
A. s  e.  X  ( ( s h 0 )  =  ( F `  s )  /\  ( s h 1 )  =  ( G `  s ) ) } )
45 ishtpy.4 . . . 4  |-  ( ph  ->  G  e.  ( J  Cn  K ) )
4623rabex 4607 . . . . 5  |-  { h  e.  ( ( J  tX  II )  Cn  K
)  |  A. s  e.  X  ( (
s h 0 )  =  ( F `  s )  /\  (
s h 1 )  =  ( G `  s ) ) }  e.  _V
4746a1i 11 . . . 4  |-  ( ph  ->  { h  e.  ( ( J  tX  II )  Cn  K )  | 
A. s  e.  X  ( ( s h 0 )  =  ( F `  s )  /\  ( s h 1 )  =  ( G `  s ) ) }  e.  _V )
4836, 44, 19, 45, 47ovmpt2d 6429 . . 3  |-  ( ph  ->  ( 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 ) ) } )
4948eleq2d 2527 . 2  |-  ( ph  ->  ( H  e.  ( F ( J Htpy  K
) G )  <->  H  e.  { h  e.  ( ( J  tX  II )  Cn  K )  | 
A. s  e.  X  ( ( s h 0 )  =  ( F `  s )  /\  ( s h 1 )  =  ( G `  s ) ) } ) )
50 oveq 6302 . . . . . 6  |-  ( h  =  H  ->  (
s h 0 )  =  ( s H 0 ) )
5150eqeq1d 2459 . . . . 5  |-  ( h  =  H  ->  (
( s h 0 )  =  ( F `
 s )  <->  ( s H 0 )  =  ( F `  s
) ) )
52 oveq 6302 . . . . . 6  |-  ( h  =  H  ->  (
s h 1 )  =  ( s H 1 ) )
5352eqeq1d 2459 . . . . 5  |-  ( h  =  H  ->  (
( s h 1 )  =  ( G `
 s )  <->  ( s H 1 )  =  ( G `  s
) ) )
5451, 53anbi12d 710 . . . 4  |-  ( h  =  H  ->  (
( ( s h 0 )  =  ( F `  s )  /\  ( s h 1 )  =  ( G `  s ) )  <->  ( ( s H 0 )  =  ( F `  s
)  /\  ( s H 1 )  =  ( G `  s
) ) ) )
5554ralbidv 2896 . . 3  |-  ( h  =  H  ->  ( A. s  e.  X  ( ( s h 0 )  =  ( F `  s )  /\  ( s h 1 )  =  ( G `  s ) )  <->  A. s  e.  X  ( ( s H 0 )  =  ( F `  s )  /\  ( s H 1 )  =  ( G `  s ) ) ) )
5655elrab 3257 . 2  |-  ( H  e.  { 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 )  /\  A. s  e.  X  ( ( s H 0 )  =  ( F `
 s )  /\  ( s H 1 )  =  ( G `
 s ) ) ) )
5749, 56syl6bb 261 1  |-  ( 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 ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1395    e. wcel 1819   A.wral 2807   {crab 2811   _Vcvv 3109    C_ wss 3471   ~Pcpw 4015   U.cuni 4251    X. cxp 5006   -->wf 5590   ` cfv 5594  (class class class)co 6296    |-> cmpt2 6298   0cc0 9509   1c1 9510   Topctop 19521  TopOnctopon 19522    Cn ccn 19852    tX ctx 20187   IIcii 21505   Htpy chtpy 21593
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6799  df-2nd 6800  df-map 7440  df-top 19526  df-topon 19529  df-cn 19855  df-htpy 21596
This theorem is referenced by:  htpycn  21599  htpyi  21600  ishtpyd  21601
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