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Theorem caoftrn 5862
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 2456 . . . . 5  |-  ( ph  ->  A. x  e.  S  A. y  e.  S  A. z  e.  S  ( ( x R y  /\  y T z )  ->  x U z ) )
32adantr 270 . . . 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 5418 . . . . 5  |-  ( (
ph  /\  w  e.  A )  ->  ( F `  w )  e.  S )
6 caofcom.3 . . . . . 6  |-  ( ph  ->  G : A --> S )
76ffvelrnda 5418 . . . . 5  |-  ( (
ph  /\  w  e.  A )  ->  ( G `  w )  e.  S )
8 caofass.4 . . . . . 6  |-  ( ph  ->  H : A --> S )
98ffvelrnda 5418 . . . . 5  |-  ( (
ph  /\  w  e.  A )  ->  ( H `  w )  e.  S )
10 breq1 3840 . . . . . . . 8  |-  ( x  =  ( F `  w )  ->  (
x R y  <->  ( F `  w ) R y ) )
1110anbi1d 453 . . . . . . 7  |-  ( x  =  ( F `  w )  ->  (
( x R y  /\  y T z )  <->  ( ( F `
 w ) R y  /\  y T z ) ) )
12 breq1 3840 . . . . . . 7  |-  ( x  =  ( F `  w )  ->  (
x U z  <->  ( F `  w ) U z ) )
1311, 12imbi12d 232 . . . . . 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 3841 . . . . . . . 8  |-  ( y  =  ( G `  w )  ->  (
( F `  w
) R y  <->  ( F `  w ) R ( G `  w ) ) )
15 breq1 3840 . . . . . . . 8  |-  ( y  =  ( G `  w )  ->  (
y T z  <->  ( G `  w ) T z ) )
1614, 15anbi12d 457 . . . . . . 7  |-  ( y  =  ( G `  w )  ->  (
( ( F `  w ) R y  /\  y T z )  <->  ( ( F `
 w ) R ( G `  w
)  /\  ( G `  w ) T z ) ) )
1716imbi1d 229 . . . . . 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 3841 . . . . . . . 8  |-  ( z  =  ( H `  w )  ->  (
( G `  w
) T z  <->  ( G `  w ) T ( H `  w ) ) )
1918anbi2d 452 . . . . . . 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 3841 . . . . . . 7  |-  ( z  =  ( H `  w )  ->  (
( F `  w
) U z  <->  ( F `  w ) U ( H `  w ) ) )
2119, 20imbi12d 232 . . . . . 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 2736 . . . . 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 1174 . . . 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 13 . . 3  |-  ( (
ph  /\  w  e.  A )  ->  (
( ( F `  w ) R ( G `  w )  /\  ( G `  w ) T ( H `  w ) )  ->  ( F `  w ) U ( H `  w ) ) )
2524ralimdva 2441 . 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 5147 . . . . . 6  |-  ( F : A --> S  ->  F  Fn  A )
274, 26syl 14 . . . . 5  |-  ( ph  ->  F  Fn  A )
28 ffn 5147 . . . . . 6  |-  ( G : A --> S  ->  G  Fn  A )
296, 28syl 14 . . . . 5  |-  ( ph  ->  G  Fn  A )
30 caofref.1 . . . . 5  |-  ( ph  ->  A  e.  V )
31 inidm 3207 . . . . 5  |-  ( A  i^i  A )  =  A
32 eqidd 2089 . . . . 5  |-  ( (
ph  /\  w  e.  A )  ->  ( F `  w )  =  ( F `  w ) )
33 eqidd 2089 . . . . 5  |-  ( (
ph  /\  w  e.  A )  ->  ( G `  w )  =  ( G `  w ) )
3427, 29, 30, 30, 31, 32, 33ofrfval 5846 . . . 4  |-  ( ph  ->  ( F  oR R G  <->  A. w  e.  A  ( F `  w ) R ( G `  w ) ) )
35 ffn 5147 . . . . . 6  |-  ( H : A --> S  ->  H  Fn  A )
368, 35syl 14 . . . . 5  |-  ( ph  ->  H  Fn  A )
37 eqidd 2089 . . . . 5  |-  ( (
ph  /\  w  e.  A )  ->  ( H `  w )  =  ( H `  w ) )
3829, 36, 30, 30, 31, 33, 37ofrfval 5846 . . . 4  |-  ( ph  ->  ( G  oR T H  <->  A. w  e.  A  ( G `  w ) T ( H `  w ) ) )
3934, 38anbi12d 457 . . 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 2497 . . 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 196 . 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 5846 . 2  |-  ( ph  ->  ( F  oR U H  <->  A. w  e.  A  ( F `  w ) U ( H `  w ) ) )
4325, 41, 423imtr4d 201 1  |-  ( ph  ->  ( ( F  oR R G  /\  G  oR T H )  ->  F  oR U H ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    /\ w3a 924    = wceq 1289    e. wcel 1438   A.wral 2359   class class class wbr 3837    Fn wfn 4997   -->wf 4998   ` cfv 5002    oRcofr 5837
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3946  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2839  df-csb 2932  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-iun 3727  df-br 3838  df-opab 3892  df-mpt 3893  df-id 4111  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009  df-fv 5010  df-ofr 5839
This theorem is referenced by: (None)
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