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Theorem caoftrn 6556
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  oR R G  /\  G  oR T H )  ->  F  oR 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 2863 . . . . 5  |-  ( ph  ->  A. x  e.  S  A. y  e.  S  A. z  e.  S  ( ( x R y  /\  y T z )  ->  x U z ) )
32adantr 465 . . . 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 6012 . . . . 5  |-  ( (
ph  /\  w  e.  A )  ->  ( F `  w )  e.  S )
6 caofcom.3 . . . . . 6  |-  ( ph  ->  G : A --> S )
76ffvelrnda 6012 . . . . 5  |-  ( (
ph  /\  w  e.  A )  ->  ( G `  w )  e.  S )
8 caofass.4 . . . . . 6  |-  ( ph  ->  H : A --> S )
98ffvelrnda 6012 . . . . 5  |-  ( (
ph  /\  w  e.  A )  ->  ( H `  w )  e.  S )
10 breq1 4436 . . . . . . . 8  |-  ( x  =  ( F `  w )  ->  (
x R y  <->  ( F `  w ) R y ) )
1110anbi1d 704 . . . . . . 7  |-  ( x  =  ( F `  w )  ->  (
( x R y  /\  y T z )  <->  ( ( F `
 w ) R y  /\  y T z ) ) )
12 breq1 4436 . . . . . . 7  |-  ( x  =  ( F `  w )  ->  (
x U z  <->  ( F `  w ) U z ) )
1311, 12imbi12d 320 . . . . . 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 4437 . . . . . . . 8  |-  ( y  =  ( G `  w )  ->  (
( F `  w
) R y  <->  ( F `  w ) R ( G `  w ) ) )
15 breq1 4436 . . . . . . . 8  |-  ( y  =  ( G `  w )  ->  (
y T z  <->  ( G `  w ) T z ) )
1614, 15anbi12d 710 . . . . . . 7  |-  ( y  =  ( G `  w )  ->  (
( ( F `  w ) R y  /\  y T z )  <->  ( ( F `
 w ) R ( G `  w
)  /\  ( G `  w ) T z ) ) )
1716imbi1d 317 . . . . . 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 4437 . . . . . . . 8  |-  ( z  =  ( H `  w )  ->  (
( G `  w
) T z  <->  ( G `  w ) T ( H `  w ) ) )
1918anbi2d 703 . . . . . . 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 4437 . . . . . . 7  |-  ( z  =  ( H `  w )  ->  (
( F `  w
) U z  <->  ( F `  w ) U ( H `  w ) ) )
2119, 20imbi12d 320 . . . . . 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 3206 . . . . 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 1227 . . . 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 2849 . 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 5717 . . . . . 6  |-  ( F : A --> S  ->  F  Fn  A )
274, 26syl 16 . . . . 5  |-  ( ph  ->  F  Fn  A )
28 ffn 5717 . . . . . 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 3689 . . . . 5  |-  ( A  i^i  A )  =  A
32 eqidd 2442 . . . . 5  |-  ( (
ph  /\  w  e.  A )  ->  ( F `  w )  =  ( F `  w ) )
33 eqidd 2442 . . . . 5  |-  ( (
ph  /\  w  e.  A )  ->  ( G `  w )  =  ( G `  w ) )
3427, 29, 30, 30, 31, 32, 33ofrfval 6529 . . . 4  |-  ( ph  ->  ( F  oR R G  <->  A. w  e.  A  ( F `  w ) R ( G `  w ) ) )
35 ffn 5717 . . . . . 6  |-  ( H : A --> S  ->  H  Fn  A )
368, 35syl 16 . . . . 5  |-  ( ph  ->  H  Fn  A )
37 eqidd 2442 . . . . 5  |-  ( (
ph  /\  w  e.  A )  ->  ( H `  w )  =  ( H `  w ) )
3829, 36, 30, 30, 31, 33, 37ofrfval 6529 . . . 4  |-  ( ph  ->  ( G  oR T H  <->  A. w  e.  A  ( G `  w ) T ( H `  w ) ) )
3934, 38anbi12d 710 . . 3  |-  ( ph  ->  ( ( F  oR R G  /\  G  oR T H )  <->  ( A. w  e.  A  ( F `  w ) R ( G `  w )  /\  A. w  e.  A  ( G `  w ) T ( H `  w ) ) ) )
40 r19.26 2968 . . 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 263 . 2  |-  ( ph  ->  ( ( F  oR R G  /\  G  oR T H )  <->  A. w  e.  A  ( ( F `  w ) R ( G `  w )  /\  ( G `  w ) T ( H `  w ) ) ) )
4227, 36, 30, 30, 31, 32, 37ofrfval 6529 . 2  |-  ( ph  ->  ( F  oR U H  <->  A. w  e.  A  ( F `  w ) U ( H `  w ) ) )
4325, 41, 423imtr4d 268 1  |-  ( ph  ->  ( ( F  oR R G  /\  G  oR T H )  ->  F  oR U H ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 972    = wceq 1381    e. wcel 1802   A.wral 2791   class class class wbr 4433    Fn wfn 5569   -->wf 5570   ` cfv 5574    oRcofr 6520
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pr 4672
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-reu 2798  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-nul 3768  df-if 3923  df-sn 4011  df-pr 4013  df-op 4017  df-uni 4231  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-id 4781  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-ofr 6522
This theorem is referenced by:  gsumbagdiaglem  17895  itg2le  22012
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