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Theorem ishtpy 18486
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 18484 . . . . . 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 10 . . . . 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 732 . . . . . . 7  |-  ( (
ph  /\  ( j  =  J  /\  k  =  K ) )  -> 
j  =  J )
4 simprr 733 . . . . . . 7  |-  ( (
ph  /\  ( j  =  J  /\  k  =  K ) )  -> 
k  =  K )
53, 4oveq12d 5892 . . . . . 6  |-  ( (
ph  /\  ( j  =  J  /\  k  =  K ) )  -> 
( j  Cn  k
)  =  ( J  Cn  K ) )
63oveq1d 5889 . . . . . . . 8  |-  ( (
ph  /\  ( j  =  J  /\  k  =  K ) )  -> 
( j  tX  II )  =  ( J  tX  II ) )
76, 4oveq12d 5892 . . . . . . 7  |-  ( (
ph  /\  ( j  =  J  /\  k  =  K ) )  -> 
( ( j  tX  II )  Cn  k
)  =  ( ( J  tX  II )  Cn  K ) )
83unieqd 3854 . . . . . . . . 9  |-  ( (
ph  /\  ( j  =  J  /\  k  =  K ) )  ->  U. j  =  U. J )
9 ishtpy.1 . . . . . . . . . . 11  |-  ( ph  ->  J  e.  (TopOn `  X ) )
10 toponuni 16681 . . . . . . . . . . 11  |-  ( J  e.  (TopOn `  X
)  ->  X  =  U. J )
119, 10syl 15 . . . . . . . . . 10  |-  ( ph  ->  X  =  U. J
)
1211adantr 451 . . . . . . . . 9  |-  ( (
ph  /\  ( j  =  J  /\  k  =  K ) )  ->  X  =  U. J )
138, 12eqtr4d 2331 . . . . . . . 8  |-  ( (
ph  /\  ( j  =  J  /\  k  =  K ) )  ->  U. j  =  X
)
1413raleqdv 2755 . . . . . . 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 2796 . . . . . 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 5926 . . . . 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 16680 . . . . . 6  |-  ( J  e.  (TopOn `  X
)  ->  J  e.  Top )
189, 17syl 15 . . . . 5  |-  ( ph  ->  J  e.  Top )
19 ishtpy.3 . . . . . 6  |-  ( ph  ->  F  e.  ( J  Cn  K ) )
20 cntop2 16987 . . . . . 6  |-  ( F  e.  ( J  Cn  K )  ->  K  e.  Top )
2119, 20syl 15 . . . . 5  |-  ( ph  ->  K  e.  Top )
22 ssrab2 3271 . . . . . . . . . 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 5899 . . . . . . . . . . 11  |-  ( ( J  tX  II )  Cn  K )  e. 
_V
2423elpw2 4191 . . . . . . . . . 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 200 . . . . . . . . 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 2624 . . . . . . . 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 2296 . . . . . . . . 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 6207 . . . . . . . 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 199 . . . . . . 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 5899 . . . . . . . 8  |-  ( J  Cn  K )  e. 
_V
3130, 30xpex 4817 . . . . . . 7  |-  ( ( J  Cn  K )  X.  ( J  Cn  K ) )  e. 
_V
3223pwex 4209 . . . . . . 7  |-  ~P (
( J  tX  II )  Cn  K )  e. 
_V
33 fex2 5417 . . . . . . 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 1277 . . . . . 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 10 . . . . 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 5991 . . . 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 5540 . . . . . . . . 9  |-  ( f  =  F  ->  (
f `  s )  =  ( F `  s ) )
3837eqeq2d 2307 . . . . . . . 8  |-  ( f  =  F  ->  (
( s h 0 )  =  ( f `
 s )  <->  ( s
h 0 )  =  ( F `  s
) ) )
39 fveq1 5540 . . . . . . . . 9  |-  ( g  =  G  ->  (
g `  s )  =  ( G `  s ) )
4039eqeq2d 2307 . . . . . . . 8  |-  ( g  =  G  ->  (
( s h 1 )  =  ( g `
 s )  <->  ( s
h 1 )  =  ( G `  s
) ) )
4138, 40bi2anan9 843 . . . . . . 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 452 . . . . . 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 2576 . . . . 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 2793 . . . 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 4181 . . . . 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 10 . . . 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 5991 . . 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 2363 . 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 5880 . . . . . 6  |-  ( h  =  H  ->  (
s h 0 )  =  ( s H 0 ) )
5150eqeq1d 2304 . . . . 5  |-  ( h  =  H  ->  (
( s h 0 )  =  ( F `
 s )  <->  ( s H 0 )  =  ( F `  s
) ) )
52 oveq 5880 . . . . . 6  |-  ( h  =  H  ->  (
s h 1 )  =  ( s H 1 ) )
5352eqeq1d 2304 . . . . 5  |-  ( h  =  H  ->  (
( s h 1 )  =  ( G `
 s )  <->  ( s H 1 )  =  ( G `  s
) ) )
5451, 53anbi12d 691 . . . 4  |-  ( h  =  H  ->  (
( ( s h 0 )  =  ( F `  s )  /\  ( s h 1 )  =  ( G `  s ) )  <->  ( ( s H 0 )  =  ( F `  s
)  /\  ( s H 1 )  =  ( G `  s
) ) ) )
5554ralbidv 2576 . . 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 2936 . 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 252 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 176    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   {crab 2560   _Vcvv 2801    C_ wss 3165   ~Pcpw 3638   U.cuni 3843    X. cxp 4703   -->wf 5267   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876   0cc0 8753   1c1 8754   Topctop 16647  TopOnctopon 16648    Cn ccn 16970    tX ctx 17271   IIcii 18395   Htpy chtpy 18481
This theorem is referenced by:  htpycn  18487  htpyi  18488  ishtpyd  18489
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-map 6790  df-top 16652  df-topon 16655  df-cn 16973  df-htpy 18484
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