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Theorem ishtpyd 21207
Description: Deduction for membership in the class of homotopies. (Contributed by Mario Carneiro, 22-Feb-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 ) )
ishtpyd.1  |-  ( ph  ->  H  e.  ( ( J  tX  II )  Cn  K ) )
ishtpyd.2  |-  ( (
ph  /\  s  e.  X )  ->  (
s H 0 )  =  ( F `  s ) )
ishtpyd.3  |-  ( (
ph  /\  s  e.  X )  ->  (
s H 1 )  =  ( G `  s ) )
Assertion
Ref Expression
ishtpyd  |-  ( ph  ->  H  e.  ( F ( J Htpy  K ) G ) )
Distinct variable groups:    F, s    G, s    H, s    J, s    ph, s    X, s
Allowed substitution hint:    K( s)

Proof of Theorem ishtpyd
StepHypRef Expression
1 ishtpyd.1 . 2  |-  ( ph  ->  H  e.  ( ( J  tX  II )  Cn  K ) )
2 ishtpyd.2 . . . 4  |-  ( (
ph  /\  s  e.  X )  ->  (
s H 0 )  =  ( F `  s ) )
3 ishtpyd.3 . . . 4  |-  ( (
ph  /\  s  e.  X )  ->  (
s H 1 )  =  ( G `  s ) )
42, 3jca 532 . . 3  |-  ( (
ph  /\  s  e.  X )  ->  (
( s H 0 )  =  ( F `
 s )  /\  ( s H 1 )  =  ( G `
 s ) ) )
54ralrimiva 2878 . 2  |-  ( ph  ->  A. s  e.  X  ( ( s H 0 )  =  ( F `  s )  /\  ( s H 1 )  =  ( G `  s ) ) )
6 ishtpy.1 . . 3  |-  ( ph  ->  J  e.  (TopOn `  X ) )
7 ishtpy.3 . . 3  |-  ( ph  ->  F  e.  ( J  Cn  K ) )
8 ishtpy.4 . . 3  |-  ( ph  ->  G  e.  ( J  Cn  K ) )
96, 7, 8ishtpy 21204 . 2  |-  ( 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 ) ) ) ) )
101, 5, 9mpbir2and 920 1  |-  ( ph  ->  H  e.  ( F ( J Htpy  K ) G ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2814   ` cfv 5586  (class class class)co 6282   0cc0 9488   1c1 9489  TopOnctopon 19159    Cn ccn 19488    tX ctx 19793   IIcii 21111   Htpy chtpy 21199
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-1st 6781  df-2nd 6782  df-map 7419  df-top 19163  df-topon 19166  df-cn 19491  df-htpy 21202
This theorem is referenced by:  htpycom  21208  htpyid  21209  htpyco1  21210  htpyco2  21211  htpycc  21212  isphtpy2d  21219
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