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Theorem ishtpyd 20547
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 2799 . 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 20544 . 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 913 1  |-  ( ph  ->  H  e.  ( F ( J Htpy  K ) G ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2715   ` cfv 5418  (class class class)co 6091   0cc0 9282   1c1 9283  TopOnctopon 18499    Cn ccn 18828    tX ctx 19133   IIcii 20451   Htpy chtpy 20539
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-fv 5426  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-1st 6577  df-2nd 6578  df-map 7216  df-top 18503  df-topon 18506  df-cn 18831  df-htpy 20542
This theorem is referenced by:  htpycom  20548  htpyid  20549  htpyco1  20550  htpyco2  20551  htpycc  20552  isphtpy2d  20559
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