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Theorem caofrss 6110
Description: Transfer a relation subset 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 )
caofrss.4  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x R y  ->  x T y ) )
Assertion
Ref Expression
caofrss  |-  ( ph  ->  ( F  o R R G  ->  F  o R T G ) )
Distinct variable groups:    x, y, F    x, G, y    ph, x, y    x, R, y    x, S, y    x, T, y
Allowed substitution hints:    A( x, y)    V( x, y)

Proof of Theorem caofrss
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 caofref.2 . . . . 5  |-  ( ph  ->  F : A --> S )
2 ffvelrn 5663 . . . . 5  |-  ( ( F : A --> S  /\  w  e.  A )  ->  ( F `  w
)  e.  S )
31, 2sylan 457 . . . 4  |-  ( (
ph  /\  w  e.  A )  ->  ( F `  w )  e.  S )
4 caofcom.3 . . . . 5  |-  ( ph  ->  G : A --> S )
5 ffvelrn 5663 . . . . 5  |-  ( ( G : A --> S  /\  w  e.  A )  ->  ( G `  w
)  e.  S )
64, 5sylan 457 . . . 4  |-  ( (
ph  /\  w  e.  A )  ->  ( G `  w )  e.  S )
7 caofrss.4 . . . . . 6  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x R y  ->  x T y ) )
87ralrimivva 2635 . . . . 5  |-  ( ph  ->  A. x  e.  S  A. y  e.  S  ( x R y  ->  x T y ) )
98adantr 451 . . . 4  |-  ( (
ph  /\  w  e.  A )  ->  A. x  e.  S  A. y  e.  S  ( x R y  ->  x T y ) )
10 breq1 4026 . . . . . 6  |-  ( x  =  ( F `  w )  ->  (
x R y  <->  ( F `  w ) R y ) )
11 breq1 4026 . . . . . 6  |-  ( x  =  ( F `  w )  ->  (
x T y  <->  ( F `  w ) T y ) )
1210, 11imbi12d 311 . . . . 5  |-  ( x  =  ( F `  w )  ->  (
( x R y  ->  x T y )  <->  ( ( F `
 w ) R y  ->  ( F `  w ) T y ) ) )
13 breq2 4027 . . . . . 6  |-  ( y  =  ( G `  w )  ->  (
( F `  w
) R y  <->  ( F `  w ) R ( G `  w ) ) )
14 breq2 4027 . . . . . 6  |-  ( y  =  ( G `  w )  ->  (
( F `  w
) T y  <->  ( F `  w ) T ( G `  w ) ) )
1513, 14imbi12d 311 . . . . 5  |-  ( y  =  ( G `  w )  ->  (
( ( F `  w ) R y  ->  ( F `  w ) T y )  <->  ( ( F `
 w ) R ( G `  w
)  ->  ( F `  w ) T ( G `  w ) ) ) )
1612, 15rspc2va 2891 . . . 4  |-  ( ( ( ( F `  w )  e.  S  /\  ( G `  w
)  e.  S )  /\  A. x  e.  S  A. y  e.  S  ( x R y  ->  x T
y ) )  -> 
( ( F `  w ) R ( G `  w )  ->  ( F `  w ) T ( G `  w ) ) )
173, 6, 9, 16syl21anc 1181 . . 3  |-  ( (
ph  /\  w  e.  A )  ->  (
( F `  w
) R ( G `
 w )  -> 
( F `  w
) T ( G `
 w ) ) )
1817ralimdva 2621 . 2  |-  ( ph  ->  ( A. w  e.  A  ( F `  w ) R ( G `  w )  ->  A. w  e.  A  ( F `  w ) T ( G `  w ) ) )
19 ffn 5389 . . . 4  |-  ( F : A --> S  ->  F  Fn  A )
201, 19syl 15 . . 3  |-  ( ph  ->  F  Fn  A )
21 ffn 5389 . . . 4  |-  ( G : A --> S  ->  G  Fn  A )
224, 21syl 15 . . 3  |-  ( ph  ->  G  Fn  A )
23 caofref.1 . . 3  |-  ( ph  ->  A  e.  V )
24 inidm 3378 . . 3  |-  ( A  i^i  A )  =  A
25 eqidd 2284 . . 3  |-  ( (
ph  /\  w  e.  A )  ->  ( F `  w )  =  ( F `  w ) )
26 eqidd 2284 . . 3  |-  ( (
ph  /\  w  e.  A )  ->  ( G `  w )  =  ( G `  w ) )
2720, 22, 23, 23, 24, 25, 26ofrfval 6086 . 2  |-  ( ph  ->  ( F  o R R G  <->  A. w  e.  A  ( F `  w ) R ( G `  w ) ) )
2820, 22, 23, 23, 24, 25, 26ofrfval 6086 . 2  |-  ( ph  ->  ( F  o R T G  <->  A. w  e.  A  ( F `  w ) T ( G `  w ) ) )
2918, 27, 283imtr4d 259 1  |-  ( ph  ->  ( F  o R R G  ->  F  o R T G ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = 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 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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