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Theorem ishtpy 18998
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 18996 . . . . . 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 734 . . . . . . 7  |-  ( (
ph  /\  ( j  =  J  /\  k  =  K ) )  -> 
j  =  J )
4 simprr 735 . . . . . . 7  |-  ( (
ph  /\  ( j  =  J  /\  k  =  K ) )  -> 
k  =  K )
53, 4oveq12d 6100 . . . . . 6  |-  ( (
ph  /\  ( j  =  J  /\  k  =  K ) )  -> 
( j  Cn  k
)  =  ( J  Cn  K ) )
63oveq1d 6097 . . . . . . . 8  |-  ( (
ph  /\  ( j  =  J  /\  k  =  K ) )  -> 
( j  tX  II )  =  ( J  tX  II ) )
76, 4oveq12d 6100 . . . . . . 7  |-  ( (
ph  /\  ( j  =  J  /\  k  =  K ) )  -> 
( ( j  tX  II )  Cn  k
)  =  ( ( J  tX  II )  Cn  K ) )
83unieqd 4027 . . . . . . . . 9  |-  ( (
ph  /\  ( j  =  J  /\  k  =  K ) )  ->  U. j  =  U. J )
9 ishtpy.1 . . . . . . . . . . 11  |-  ( ph  ->  J  e.  (TopOn `  X ) )
10 toponuni 16993 . . . . . . . . . . 11  |-  ( J  e.  (TopOn `  X
)  ->  X  =  U. J )
119, 10syl 16 . . . . . . . . . 10  |-  ( ph  ->  X  =  U. J
)
1211adantr 453 . . . . . . . . 9  |-  ( (
ph  /\  ( j  =  J  /\  k  =  K ) )  ->  X  =  U. J )
138, 12eqtr4d 2472 . . . . . . . 8  |-  ( (
ph  /\  ( j  =  J  /\  k  =  K ) )  ->  U. j  =  X
)
1413raleqdv 2911 . . . . . . 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 2952 . . . . . 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 6137 . . . . 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 16992 . . . . . 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 17306 . . . . . 6  |-  ( F  e.  ( J  Cn  K )  ->  K  e.  Top )
2119, 20syl 16 . . . . 5  |-  ( ph  ->  K  e.  Top )
22 ssrab2 3429 . . . . . . . . . 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 6107 . . . . . . . . . . 11  |-  ( ( J  tX  II )  Cn  K )  e. 
_V
2423elpw2 4365 . . . . . . . . . 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 202 . . . . . . . . 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 2775 . . . . . . . 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 2437 . . . . . . . . 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 6419 . . . . . . . 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 201 . . . . . . 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 6107 . . . . . . . 8  |-  ( J  Cn  K )  e. 
_V
3130, 30xpex 4991 . . . . . . 7  |-  ( ( J  Cn  K )  X.  ( J  Cn  K ) )  e. 
_V
3223pwex 4383 . . . . . . 7  |-  ~P (
( J  tX  II )  Cn  K )  e. 
_V
33 fex2 5604 . . . . . . 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 1280 . . . . . 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 6202 . . . 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 5728 . . . . . . . . 9  |-  ( f  =  F  ->  (
f `  s )  =  ( F `  s ) )
3837eqeq2d 2448 . . . . . . . 8  |-  ( f  =  F  ->  (
( s h 0 )  =  ( f `
 s )  <->  ( s
h 0 )  =  ( F `  s
) ) )
39 fveq1 5728 . . . . . . . . 9  |-  ( g  =  G  ->  (
g `  s )  =  ( G `  s ) )
4039eqeq2d 2448 . . . . . . . 8  |-  ( g  =  G  ->  (
( s h 1 )  =  ( g `
 s )  <->  ( s
h 1 )  =  ( G `  s
) ) )
4138, 40bi2anan9 845 . . . . . . 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 454 . . . . . 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 2726 . . . . 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 2949 . . . 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 4355 . . . . 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 6202 . . 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 2504 . 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 6088 . . . . . 6  |-  ( h  =  H  ->  (
s h 0 )  =  ( s H 0 ) )
5150eqeq1d 2445 . . . . 5  |-  ( h  =  H  ->  (
( s h 0 )  =  ( F `
 s )  <->  ( s H 0 )  =  ( F `  s
) ) )
52 oveq 6088 . . . . . 6  |-  ( h  =  H  ->  (
s h 1 )  =  ( s H 1 ) )
5352eqeq1d 2445 . . . . 5  |-  ( h  =  H  ->  (
( s h 1 )  =  ( G `
 s )  <->  ( s H 1 )  =  ( G `  s
) ) )
5451, 53anbi12d 693 . . . 4  |-  ( h  =  H  ->  (
( ( s h 0 )  =  ( F `  s )  /\  ( s h 1 )  =  ( G `  s ) )  <->  ( ( s H 0 )  =  ( F `  s
)  /\  ( s H 1 )  =  ( G `  s
) ) ) )
5554ralbidv 2726 . . 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 3093 . 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 254 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 set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2706   {crab 2710   _Vcvv 2957    C_ wss 3321   ~Pcpw 3800   U.cuni 4016    X. cxp 4877   -->wf 5451   ` cfv 5455  (class class class)co 6082    e. cmpt2 6084   0cc0 8991   1c1 8992   Topctop 16959  TopOnctopon 16960    Cn ccn 17289    tX ctx 17593   IIcii 18906   Htpy chtpy 18993
This theorem is referenced by:  htpycn  18999  htpyi  19000  ishtpyd  19001
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-1st 6350  df-2nd 6351  df-map 7021  df-top 16964  df-topon 16967  df-cn 17292  df-htpy 18996
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