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Theorem caoftrn 6128
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 2649 . . . . 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 5679 . . . . . 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 5679 . . . . . 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 5679 . . . . . 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 4042 . . . . . . . 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 4042 . . . . . . 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 4043 . . . . . . . 8  |-  ( y  =  ( G `  w )  ->  (
( F `  w
) R y  <->  ( F `  w ) R ( G `  w ) ) )
18 breq1 4042 . . . . . . . 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 4043 . . . . . . . 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 4043 . . . . . . 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 2906 . . . . 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 2634 . 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 5405 . . . . . 6  |-  ( F : A --> S  ->  F  Fn  A )
304, 29syl 15 . . . . 5  |-  ( ph  ->  F  Fn  A )
31 ffn 5405 . . . . . 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 3391 . . . . 5  |-  ( A  i^i  A )  =  A
35 eqidd 2297 . . . . 5  |-  ( (
ph  /\  w  e.  A )  ->  ( F `  w )  =  ( F `  w ) )
36 eqidd 2297 . . . . 5  |-  ( (
ph  /\  w  e.  A )  ->  ( G `  w )  =  ( G `  w ) )
3730, 32, 33, 33, 34, 35, 36ofrfval 6102 . . . 4  |-  ( ph  ->  ( F  o R R G  <->  A. w  e.  A  ( F `  w ) R ( G `  w ) ) )
38 ffn 5405 . . . . . 6  |-  ( H : A --> S  ->  H  Fn  A )
3910, 38syl 15 . . . . 5  |-  ( ph  ->  H  Fn  A )
40 eqidd 2297 . . . . 5  |-  ( (
ph  /\  w  e.  A )  ->  ( H `  w )  =  ( H `  w ) )
4132, 39, 33, 33, 34, 36, 40ofrfval 6102 . . . 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 2688 . . 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 6102 . 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 1632    e. wcel 1696   A.wral 2556   class class class wbr 4039    Fn wfn 5266   -->wf 5267   ` cfv 5271    o Rcofr 6093
This theorem is referenced by:  gsumbagdiaglem  16137  itg2le  19110
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-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pr 4230
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-reu 2563  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-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-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ofr 6095
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