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Theorem ishtpy 22015
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 22013 . . . . . 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 765 . . . . . . 7  |-  ( (
ph  /\  ( j  =  J  /\  k  =  K ) )  -> 
j  =  J )
4 simprr 767 . . . . . . 7  |-  ( (
ph  /\  ( j  =  J  /\  k  =  K ) )  -> 
k  =  K )
53, 4oveq12d 6313 . . . . . 6  |-  ( (
ph  /\  ( j  =  J  /\  k  =  K ) )  -> 
( j  Cn  k
)  =  ( J  Cn  K ) )
63oveq1d 6310 . . . . . . . 8  |-  ( (
ph  /\  ( j  =  J  /\  k  =  K ) )  -> 
( j  tX  II )  =  ( J  tX  II ) )
76, 4oveq12d 6313 . . . . . . 7  |-  ( (
ph  /\  ( j  =  J  /\  k  =  K ) )  -> 
( ( j  tX  II )  Cn  k
)  =  ( ( J  tX  II )  Cn  K ) )
83unieqd 4211 . . . . . . . . 9  |-  ( (
ph  /\  ( j  =  J  /\  k  =  K ) )  ->  U. j  =  U. J )
9 ishtpy.1 . . . . . . . . . . 11  |-  ( ph  ->  J  e.  (TopOn `  X ) )
10 toponuni 19954 . . . . . . . . . . 11  |-  ( J  e.  (TopOn `  X
)  ->  X  =  U. J )
119, 10syl 17 . . . . . . . . . 10  |-  ( ph  ->  X  =  U. J
)
1211adantr 467 . . . . . . . . 9  |-  ( (
ph  /\  ( j  =  J  /\  k  =  K ) )  ->  X  =  U. J )
138, 12eqtr4d 2490 . . . . . . . 8  |-  ( (
ph  /\  ( j  =  J  /\  k  =  K ) )  ->  U. j  =  X
)
1413raleqdv 2995 . . . . . . 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 3042 . . . . . 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 19953 . . . . . 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 20269 . . . . . 6  |-  ( F  e.  ( J  Cn  K )  ->  K  e.  Top )
2119, 20syl 17 . . . . 5  |-  ( ph  ->  K  e.  Top )
22 ssrab2 3516 . . . . . . . . . 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 6323 . . . . . . . . . . 11  |-  ( ( J  tX  II )  Cn  K )  e. 
_V
2423elpw2 4570 . . . . . . . . . 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 213 . . . . . . . . 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 2752 . . . . . . . 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 2453 . . . . . . . . 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 6865 . . . . . . . 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 212 . . . . . . 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 6323 . . . . . . . 8  |-  ( J  Cn  K )  e. 
_V
3130, 30xpex 6600 . . . . . . 7  |-  ( ( J  Cn  K )  X.  ( J  Cn  K ) )  e. 
_V
3223pwex 4589 . . . . . . 7  |-  ~P (
( J  tX  II )  Cn  K )  e. 
_V
33 fex2 6753 . . . . . . 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 1366 . . . . . 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 5869 . . . . . . . . 9  |-  ( f  =  F  ->  (
f `  s )  =  ( F `  s ) )
3837eqeq2d 2463 . . . . . . . 8  |-  ( f  =  F  ->  (
( s h 0 )  =  ( f `
 s )  <->  ( s
h 0 )  =  ( F `  s
) ) )
39 fveq1 5869 . . . . . . . . 9  |-  ( g  =  G  ->  (
g `  s )  =  ( G `  s ) )
4039eqeq2d 2463 . . . . . . . 8  |-  ( g  =  G  ->  (
( s h 1 )  =  ( g `
 s )  <->  ( s
h 1 )  =  ( G `  s
) ) )
4138, 40bi2anan9 885 . . . . . . 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 468 . . . . . 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 3038 . . . 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 4557 . . . . 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 2516 . 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 6301 . . . . . 6  |-  ( h  =  H  ->  (
s h 0 )  =  ( s H 0 ) )
5150eqeq1d 2455 . . . . 5  |-  ( h  =  H  ->  (
( s h 0 )  =  ( F `
 s )  <->  ( s H 0 )  =  ( F `  s
) ) )
52 oveq 6301 . . . . . 6  |-  ( h  =  H  ->  (
s h 1 )  =  ( s H 1 ) )
5352eqeq1d 2455 . . . . 5  |-  ( h  =  H  ->  (
( s h 1 )  =  ( G `
 s )  <->  ( s H 1 )  =  ( G `  s
) ) )
5451, 53anbi12d 718 . . . 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 3198 . 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 265 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 188    /\ wa 371    = wceq 1446    e. wcel 1889   A.wral 2739   {crab 2743   _Vcvv 3047    C_ wss 3406   ~Pcpw 3953   U.cuni 4201    X. cxp 4835   -->wf 5581   ` cfv 5585  (class class class)co 6295    |-> cmpt2 6297   0cc0 9544   1c1 9545   Topctop 19929  TopOnctopon 19930    Cn ccn 20252    tX ctx 20587   IIcii 21919   Htpy chtpy 22010
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-ral 2744  df-rex 2745  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-op 3977  df-uni 4202  df-iun 4283  df-br 4406  df-opab 4465  df-mpt 4466  df-id 4752  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-fv 5593  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-1st 6798  df-2nd 6799  df-map 7479  df-top 19933  df-topon 19935  df-cn 20255  df-htpy 22013
This theorem is referenced by:  htpycn  22016  htpyi  22017  ishtpyd  22018
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