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Theorem ishtpyd 18872
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 519 . . 3  |-  ( (
ph  /\  s  e.  X )  ->  (
( s H 0 )  =  ( F `
 s )  /\  ( s H 1 )  =  ( G `
 s ) ) )
54ralrimiva 2733 . 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 18869 . 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 889 1  |-  ( ph  ->  H  e.  ( F ( J Htpy  K ) G ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   A.wral 2650   ` cfv 5395  (class class class)co 6021   0cc0 8924   1c1 8925  TopOnctopon 16883    Cn ccn 17211    tX ctx 17514   IIcii 18777   Htpy chtpy 18864
This theorem is referenced by:  htpycom  18873  htpyid  18874  htpyco1  18875  htpyco2  18876  htpycc  18877  isphtpy2d  18884
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-1st 6289  df-2nd 6290  df-map 6957  df-top 16887  df-topon 16890  df-cn 17214  df-htpy 18867
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