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Theorem ishtpy 22081
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 22079 . . . . . 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 772 . . . . . . 7  |-  ( (
ph  /\  ( j  =  J  /\  k  =  K ) )  -> 
j  =  J )
4 simprr 774 . . . . . . 7  |-  ( (
ph  /\  ( j  =  J  /\  k  =  K ) )  -> 
k  =  K )
53, 4oveq12d 6326 . . . . . 6  |-  ( (
ph  /\  ( j  =  J  /\  k  =  K ) )  -> 
( j  Cn  k
)  =  ( J  Cn  K ) )
63oveq1d 6323 . . . . . . . 8  |-  ( (
ph  /\  ( j  =  J  /\  k  =  K ) )  -> 
( j  tX  II )  =  ( J  tX  II ) )
76, 4oveq12d 6326 . . . . . . 7  |-  ( (
ph  /\  ( j  =  J  /\  k  =  K ) )  -> 
( ( j  tX  II )  Cn  k
)  =  ( ( J  tX  II )  Cn  K ) )
83unieqd 4200 . . . . . . . . 9  |-  ( (
ph  /\  ( j  =  J  /\  k  =  K ) )  ->  U. j  =  U. J )
9 ishtpy.1 . . . . . . . . . . 11  |-  ( ph  ->  J  e.  (TopOn `  X ) )
10 toponuni 20019 . . . . . . . . . . 11  |-  ( J  e.  (TopOn `  X
)  ->  X  =  U. J )
119, 10syl 17 . . . . . . . . . 10  |-  ( ph  ->  X  =  U. J
)
1211adantr 472 . . . . . . . . 9  |-  ( (
ph  /\  ( j  =  J  /\  k  =  K ) )  ->  X  =  U. J )
138, 12eqtr4d 2508 . . . . . . . 8  |-  ( (
ph  /\  ( j  =  J  /\  k  =  K ) )  ->  U. j  =  X
)
1413raleqdv 2979 . . . . . . 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 3026 . . . . . 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 6372 . . . . 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 20018 . . . . . 6  |-  ( J  e.  (TopOn `  X
)  ->  J  e.  Top )
189, 17syl 17 . . . . 5  |-  ( ph  ->  J  e.  Top )
19 ishtpy.3 . . . . . 6  |-  ( ph  ->  F  e.  ( J  Cn  K ) )
20 cntop2 20334 . . . . . 6  |-  ( F  e.  ( J  Cn  K )  ->  K  e.  Top )
2119, 20syl 17 . . . . 5  |-  ( ph  ->  K  e.  Top )
22 ssrab2 3500 . . . . . . . . . 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 6336 . . . . . . . . . . 11  |-  ( ( J  tX  II )  Cn  K )  e. 
_V
2423elpw2 4565 . . . . . . . . . 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 214 . . . . . . . . 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 2769 . . . . . . . 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 2471 . . . . . . . . 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 6879 . . . . . . . 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 213 . . . . . . 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 6336 . . . . . . . 8  |-  ( J  Cn  K )  e. 
_V
3130, 30xpex 6614 . . . . . . 7  |-  ( ( J  Cn  K )  X.  ( J  Cn  K ) )  e. 
_V
3223pwex 4584 . . . . . . 7  |-  ~P (
( J  tX  II )  Cn  K )  e. 
_V
33 fex2 6767 . . . . . . 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 1390 . . . . . 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 6443 . . . 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 5878 . . . . . . . . 9  |-  ( f  =  F  ->  (
f `  s )  =  ( F `  s ) )
3837eqeq2d 2481 . . . . . . . 8  |-  ( f  =  F  ->  (
( s h 0 )  =  ( f `
 s )  <->  ( s
h 0 )  =  ( F `  s
) ) )
39 fveq1 5878 . . . . . . . . 9  |-  ( g  =  G  ->  (
g `  s )  =  ( G `  s ) )
4039eqeq2d 2481 . . . . . . . 8  |-  ( g  =  G  ->  (
( s h 1 )  =  ( g `
 s )  <->  ( s
h 1 )  =  ( G `  s
) ) )
4138, 40bi2anan9 890 . . . . . . 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 473 . . . . . 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 2829 . . . . 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 3022 . . . 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 4550 . . . . 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 6443 . . 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 2534 . 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 6314 . . . . . 6  |-  ( h  =  H  ->  (
s h 0 )  =  ( s H 0 ) )
5150eqeq1d 2473 . . . . 5  |-  ( h  =  H  ->  (
( s h 0 )  =  ( F `
 s )  <->  ( s H 0 )  =  ( F `  s
) ) )
52 oveq 6314 . . . . . 6  |-  ( h  =  H  ->  (
s h 1 )  =  ( s H 1 ) )
5352eqeq1d 2473 . . . . 5  |-  ( h  =  H  ->  (
( s h 1 )  =  ( G `
 s )  <->  ( s H 1 )  =  ( G `  s
) ) )
5451, 53anbi12d 725 . . . 4  |-  ( h  =  H  ->  (
( ( s h 0 )  =  ( F `  s )  /\  ( s h 1 )  =  ( G `  s ) )  <->  ( ( s H 0 )  =  ( F `  s
)  /\  ( s H 1 )  =  ( G `  s
) ) ) )
5554ralbidv 2829 . . 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 3184 . 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 269 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 189    /\ wa 376    = wceq 1452    e. wcel 1904   A.wral 2756   {crab 2760   _Vcvv 3031    C_ wss 3390   ~Pcpw 3942   U.cuni 4190    X. cxp 4837   -->wf 5585   ` cfv 5589  (class class class)co 6308    |-> cmpt2 6310   0cc0 9557   1c1 9558   Topctop 19994  TopOnctopon 19995    Cn ccn 20317    tX ctx 20652   IIcii 21985   Htpy chtpy 22076
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-fv 5597  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-1st 6812  df-2nd 6813  df-map 7492  df-top 19998  df-topon 20000  df-cn 20320  df-htpy 22079
This theorem is referenced by:  htpycn  22082  htpyi  22083  ishtpyd  22084
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