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Theorem caoftrn 6339
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 2799 . . . . 5  |-  ( ph  ->  A. x  e.  S  A. y  e.  S  A. z  e.  S  ( ( x R y  /\  y T z )  ->  x U z ) )
32adantr 452 . . . 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 )
54ffvelrnda 5870 . . . . 5  |-  ( (
ph  /\  w  e.  A )  ->  ( F `  w )  e.  S )
6 caofcom.3 . . . . . 6  |-  ( ph  ->  G : A --> S )
76ffvelrnda 5870 . . . . 5  |-  ( (
ph  /\  w  e.  A )  ->  ( G `  w )  e.  S )
8 caofass.4 . . . . . 6  |-  ( ph  ->  H : A --> S )
98ffvelrnda 5870 . . . . 5  |-  ( (
ph  /\  w  e.  A )  ->  ( H `  w )  e.  S )
10 breq1 4215 . . . . . . . 8  |-  ( x  =  ( F `  w )  ->  (
x R y  <->  ( F `  w ) R y ) )
1110anbi1d 686 . . . . . . 7  |-  ( x  =  ( F `  w )  ->  (
( x R y  /\  y T z )  <->  ( ( F `
 w ) R y  /\  y T z ) ) )
12 breq1 4215 . . . . . . 7  |-  ( x  =  ( F `  w )  ->  (
x U z  <->  ( F `  w ) U z ) )
1311, 12imbi12d 312 . . . . . 6  |-  ( x  =  ( F `  w )  ->  (
( ( x R y  /\  y T z )  ->  x U z )  <->  ( (
( F `  w
) R y  /\  y T z )  -> 
( F `  w
) U z ) ) )
14 breq2 4216 . . . . . . . 8  |-  ( y  =  ( G `  w )  ->  (
( F `  w
) R y  <->  ( F `  w ) R ( G `  w ) ) )
15 breq1 4215 . . . . . . . 8  |-  ( y  =  ( G `  w )  ->  (
y T z  <->  ( G `  w ) T z ) )
1614, 15anbi12d 692 . . . . . . 7  |-  ( y  =  ( G `  w )  ->  (
( ( F `  w ) R y  /\  y T z )  <->  ( ( F `
 w ) R ( G `  w
)  /\  ( G `  w ) T z ) ) )
1716imbi1d 309 . . . . . 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 ) ) )
18 breq2 4216 . . . . . . . 8  |-  ( z  =  ( H `  w )  ->  (
( G `  w
) T z  <->  ( G `  w ) T ( H `  w ) ) )
1918anbi2d 685 . . . . . . 7  |-  ( z  =  ( H `  w )  ->  (
( ( F `  w ) R ( G `  w )  /\  ( G `  w ) T z )  <->  ( ( F `
 w ) R ( G `  w
)  /\  ( G `  w ) T ( H `  w ) ) ) )
20 breq2 4216 . . . . . . 7  |-  ( z  =  ( H `  w )  ->  (
( F `  w
) U z  <->  ( F `  w ) U ( H `  w ) ) )
2119, 20imbi12d 312 . . . . . 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 ) ) ) )
2213, 17, 21rspc3v 3061 . . . . 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 ) ) ) )
235, 7, 9, 22syl3anc 1184 . . . 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 ) ) ) )
243, 23mpd 15 . . 3  |-  ( (
ph  /\  w  e.  A )  ->  (
( ( F `  w ) R ( G `  w )  /\  ( G `  w ) T ( H `  w ) )  ->  ( F `  w ) U ( H `  w ) ) )
2524ralimdva 2784 . 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 ) ) )
26 ffn 5591 . . . . . 6  |-  ( F : A --> S  ->  F  Fn  A )
274, 26syl 16 . . . . 5  |-  ( ph  ->  F  Fn  A )
28 ffn 5591 . . . . . 6  |-  ( G : A --> S  ->  G  Fn  A )
296, 28syl 16 . . . . 5  |-  ( ph  ->  G  Fn  A )
30 caofref.1 . . . . 5  |-  ( ph  ->  A  e.  V )
31 inidm 3550 . . . . 5  |-  ( A  i^i  A )  =  A
32 eqidd 2437 . . . . 5  |-  ( (
ph  /\  w  e.  A )  ->  ( F `  w )  =  ( F `  w ) )
33 eqidd 2437 . . . . 5  |-  ( (
ph  /\  w  e.  A )  ->  ( G `  w )  =  ( G `  w ) )
3427, 29, 30, 30, 31, 32, 33ofrfval 6313 . . . 4  |-  ( ph  ->  ( F  o R R G  <->  A. w  e.  A  ( F `  w ) R ( G `  w ) ) )
35 ffn 5591 . . . . . 6  |-  ( H : A --> S  ->  H  Fn  A )
368, 35syl 16 . . . . 5  |-  ( ph  ->  H  Fn  A )
37 eqidd 2437 . . . . 5  |-  ( (
ph  /\  w  e.  A )  ->  ( H `  w )  =  ( H `  w ) )
3829, 36, 30, 30, 31, 33, 37ofrfval 6313 . . . 4  |-  ( ph  ->  ( G  o R T H  <->  A. w  e.  A  ( G `  w ) T ( H `  w ) ) )
3934, 38anbi12d 692 . . 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 ) ) ) )
40 r19.26 2838 . . 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 ) ) )
4139, 40syl6bbr 255 . 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 ) ) ) )
4227, 36, 30, 30, 31, 32, 37ofrfval 6313 . 2  |-  ( ph  ->  ( F  o R U H  <->  A. w  e.  A  ( F `  w ) U ( H `  w ) ) )
4325, 41, 423imtr4d 260 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 359    /\ w3a 936    = wceq 1652    e. wcel 1725   A.wral 2705   class class class wbr 4212    Fn wfn 5449   -->wf 5450   ` cfv 5454    o Rcofr 6304
This theorem is referenced by:  gsumbagdiaglem  16440  itg2le  19631
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ofr 6306
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