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Theorem caoftrn 6112
Description: Transfer a transitivity law to the function relation. (Contributed by Mario Carneiro, 28-Jul-2014.)
Hypotheses
Ref Expression
caofref.1  |-  ( ph  ->  A  e.  V )
caofref.2  |-  ( ph  ->  F : A --> S )
caofcom.3  |-  ( ph  ->  G : A --> S )
caofass.4  |-  ( ph  ->  H : A --> S )
caoftrn.5  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S  /\  z  e.  S ) )  -> 
( ( x R y  /\  y T z )  ->  x U z ) )
Assertion
Ref Expression
caoftrn  |-  ( ph  ->  ( ( F  o R R G  /\  G  o R T H )  ->  F  o R U H ) )
Distinct variable groups:    x, y,
z, F    x, G, y, z    x, H, y, z    ph, x, y, z   
x, R, y, z   
x, S, y, z   
x, T, y, z   
x, U, y, z
Allowed substitution hints:    A( x, y, z)    V( x, y, z)

Proof of Theorem caoftrn
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 caoftrn.5 . . . . . 6  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S  /\  z  e.  S ) )  -> 
( ( x R y  /\  y T z )  ->  x U z ) )
21ralrimivvva 2636 . . . . 5  |-  ( ph  ->  A. x  e.  S  A. y  e.  S  A. z  e.  S  ( ( x R y  /\  y T z )  ->  x U z ) )
32adantr 451 . . . 4  |-  ( (
ph  /\  w  e.  A )  ->  A. x  e.  S  A. y  e.  S  A. z  e.  S  ( (
x R y  /\  y T z )  ->  x U z ) )
4 caofref.2 . . . . . 6  |-  ( ph  ->  F : A --> S )
5 ffvelrn 5663 . . . . . 6  |-  ( ( F : A --> S  /\  w  e.  A )  ->  ( F `  w
)  e.  S )
64, 5sylan 457 . . . . 5  |-  ( (
ph  /\  w  e.  A )  ->  ( F `  w )  e.  S )
7 caofcom.3 . . . . . 6  |-  ( ph  ->  G : A --> S )
8 ffvelrn 5663 . . . . . 6  |-  ( ( G : A --> S  /\  w  e.  A )  ->  ( G `  w
)  e.  S )
97, 8sylan 457 . . . . 5  |-  ( (
ph  /\  w  e.  A )  ->  ( G `  w )  e.  S )
10 caofass.4 . . . . . 6  |-  ( ph  ->  H : A --> S )
11 ffvelrn 5663 . . . . . 6  |-  ( ( H : A --> S  /\  w  e.  A )  ->  ( H `  w
)  e.  S )
1210, 11sylan 457 . . . . 5  |-  ( (
ph  /\  w  e.  A )  ->  ( H `  w )  e.  S )
13 breq1 4026 . . . . . . . 8  |-  ( x  =  ( F `  w )  ->  (
x R y  <->  ( F `  w ) R y ) )
1413anbi1d 685 . . . . . . 7  |-  ( x  =  ( F `  w )  ->  (
( x R y  /\  y T z )  <->  ( ( F `
 w ) R y  /\  y T z ) ) )
15 breq1 4026 . . . . . . 7  |-  ( x  =  ( F `  w )  ->  (
x U z  <->  ( F `  w ) U z ) )
1614, 15imbi12d 311 . . . . . 6  |-  ( x  =  ( F `  w )  ->  (
( ( x R y  /\  y T z )  ->  x U z )  <->  ( (
( F `  w
) R y  /\  y T z )  -> 
( F `  w
) U z ) ) )
17 breq2 4027 . . . . . . . 8  |-  ( y  =  ( G `  w )  ->  (
( F `  w
) R y  <->  ( F `  w ) R ( G `  w ) ) )
18 breq1 4026 . . . . . . . 8  |-  ( y  =  ( G `  w )  ->  (
y T z  <->  ( G `  w ) T z ) )
1917, 18anbi12d 691 . . . . . . 7  |-  ( y  =  ( G `  w )  ->  (
( ( F `  w ) R y  /\  y T z )  <->  ( ( F `
 w ) R ( G `  w
)  /\  ( G `  w ) T z ) ) )
2019imbi1d 308 . . . . . 6  |-  ( y  =  ( G `  w )  ->  (
( ( ( F `
 w ) R y  /\  y T z )  ->  ( F `  w ) U z )  <->  ( (
( F `  w
) R ( G `
 w )  /\  ( G `  w ) T z )  -> 
( F `  w
) U z ) ) )
21 breq2 4027 . . . . . . . 8  |-  ( z  =  ( H `  w )  ->  (
( G `  w
) T z  <->  ( G `  w ) T ( H `  w ) ) )
2221anbi2d 684 . . . . . . 7  |-  ( z  =  ( H `  w )  ->  (
( ( F `  w ) R ( G `  w )  /\  ( G `  w ) T z )  <->  ( ( F `
 w ) R ( G `  w
)  /\  ( G `  w ) T ( H `  w ) ) ) )
23 breq2 4027 . . . . . . 7  |-  ( z  =  ( H `  w )  ->  (
( F `  w
) U z  <->  ( F `  w ) U ( H `  w ) ) )
2422, 23imbi12d 311 . . . . . 6  |-  ( z  =  ( H `  w )  ->  (
( ( ( F `
 w ) R ( G `  w
)  /\  ( G `  w ) T z )  ->  ( F `  w ) U z )  <->  ( ( ( F `  w ) R ( G `  w )  /\  ( G `  w ) T ( H `  w ) )  -> 
( F `  w
) U ( H `
 w ) ) ) )
2516, 20, 24rspc3v 2893 . . . . 5  |-  ( ( ( F `  w
)  e.  S  /\  ( G `  w )  e.  S  /\  ( H `  w )  e.  S )  ->  ( A. x  e.  S  A. y  e.  S  A. z  e.  S  ( ( x R y  /\  y T z )  ->  x U z )  -> 
( ( ( F `
 w ) R ( G `  w
)  /\  ( G `  w ) T ( H `  w ) )  ->  ( F `  w ) U ( H `  w ) ) ) )
266, 9, 12, 25syl3anc 1182 . . . 4  |-  ( (
ph  /\  w  e.  A )  ->  ( A. x  e.  S  A. y  e.  S  A. z  e.  S  ( ( x R y  /\  y T z )  ->  x U z )  -> 
( ( ( F `
 w ) R ( G `  w
)  /\  ( G `  w ) T ( H `  w ) )  ->  ( F `  w ) U ( H `  w ) ) ) )
273, 26mpd 14 . . 3  |-  ( (
ph  /\  w  e.  A )  ->  (
( ( F `  w ) R ( G `  w )  /\  ( G `  w ) T ( H `  w ) )  ->  ( F `  w ) U ( H `  w ) ) )
2827ralimdva 2621 . 2  |-  ( ph  ->  ( A. w  e.  A  ( ( F `
 w ) R ( G `  w
)  /\  ( G `  w ) T ( H `  w ) )  ->  A. w  e.  A  ( F `  w ) U ( H `  w ) ) )
29 ffn 5389 . . . . . 6  |-  ( F : A --> S  ->  F  Fn  A )
304, 29syl 15 . . . . 5  |-  ( ph  ->  F  Fn  A )
31 ffn 5389 . . . . . 6  |-  ( G : A --> S  ->  G  Fn  A )
327, 31syl 15 . . . . 5  |-  ( ph  ->  G  Fn  A )
33 caofref.1 . . . . 5  |-  ( ph  ->  A  e.  V )
34 inidm 3378 . . . . 5  |-  ( A  i^i  A )  =  A
35 eqidd 2284 . . . . 5  |-  ( (
ph  /\  w  e.  A )  ->  ( F `  w )  =  ( F `  w ) )
36 eqidd 2284 . . . . 5  |-  ( (
ph  /\  w  e.  A )  ->  ( G `  w )  =  ( G `  w ) )
3730, 32, 33, 33, 34, 35, 36ofrfval 6086 . . . 4  |-  ( ph  ->  ( F  o R R G  <->  A. w  e.  A  ( F `  w ) R ( G `  w ) ) )
38 ffn 5389 . . . . . 6  |-  ( H : A --> S  ->  H  Fn  A )
3910, 38syl 15 . . . . 5  |-  ( ph  ->  H  Fn  A )
40 eqidd 2284 . . . . 5  |-  ( (
ph  /\  w  e.  A )  ->  ( H `  w )  =  ( H `  w ) )
4132, 39, 33, 33, 34, 36, 40ofrfval 6086 . . . 4  |-  ( ph  ->  ( G  o R T H  <->  A. w  e.  A  ( G `  w ) T ( H `  w ) ) )
4237, 41anbi12d 691 . . 3  |-  ( ph  ->  ( ( F  o R R G  /\  G  o R T H )  <-> 
( A. w  e.  A  ( F `  w ) R ( G `  w )  /\  A. w  e.  A  ( G `  w ) T ( H `  w ) ) ) )
43 r19.26 2675 . . 3  |-  ( A. w  e.  A  (
( F `  w
) R ( G `
 w )  /\  ( G `  w ) T ( H `  w ) )  <->  ( A. w  e.  A  ( F `  w ) R ( G `  w )  /\  A. w  e.  A  ( G `  w ) T ( H `  w ) ) )
4442, 43syl6bbr 254 . 2  |-  ( ph  ->  ( ( F  o R R G  /\  G  o R T H )  <->  A. w  e.  A  ( ( F `  w ) R ( G `  w )  /\  ( G `  w ) T ( H `  w ) ) ) )
4530, 39, 33, 33, 34, 35, 40ofrfval 6086 . 2  |-  ( ph  ->  ( F  o R U H  <->  A. w  e.  A  ( F `  w ) U ( H `  w ) ) )
4628, 44, 453imtr4d 259 1  |-  ( ph  ->  ( ( F  o R R G  /\  G  o R T H )  ->  F  o R U H ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   class class class wbr 4023    Fn wfn 5250   -->wf 5251   ` cfv 5255    o Rcofr 6077
This theorem is referenced by:  gsumbagdiaglem  16121  itg2le  19094
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ofr 6079
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