MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  caofrss Structured version   Unicode version

Theorem caofrss 6353
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  oR R G  ->  F  oR 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 )
21ffvelrnda 5843 . . . 4  |-  ( (
ph  /\  w  e.  A )  ->  ( F `  w )  e.  S )
3 caofcom.3 . . . . 5  |-  ( ph  ->  G : A --> S )
43ffvelrnda 5843 . . . 4  |-  ( (
ph  /\  w  e.  A )  ->  ( G `  w )  e.  S )
5 caofrss.4 . . . . . 6  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x R y  ->  x T y ) )
65ralrimivva 2808 . . . . 5  |-  ( ph  ->  A. x  e.  S  A. y  e.  S  ( x R y  ->  x T y ) )
76adantr 465 . . . 4  |-  ( (
ph  /\  w  e.  A )  ->  A. x  e.  S  A. y  e.  S  ( x R y  ->  x T y ) )
8 breq1 4295 . . . . . 6  |-  ( x  =  ( F `  w )  ->  (
x R y  <->  ( F `  w ) R y ) )
9 breq1 4295 . . . . . 6  |-  ( x  =  ( F `  w )  ->  (
x T y  <->  ( F `  w ) T y ) )
108, 9imbi12d 320 . . . . 5  |-  ( x  =  ( F `  w )  ->  (
( x R y  ->  x T y )  <->  ( ( F `
 w ) R y  ->  ( F `  w ) T y ) ) )
11 breq2 4296 . . . . . 6  |-  ( y  =  ( G `  w )  ->  (
( F `  w
) R y  <->  ( F `  w ) R ( G `  w ) ) )
12 breq2 4296 . . . . . 6  |-  ( y  =  ( G `  w )  ->  (
( F `  w
) T y  <->  ( F `  w ) T ( G `  w ) ) )
1311, 12imbi12d 320 . . . . 5  |-  ( y  =  ( G `  w )  ->  (
( ( F `  w ) R y  ->  ( F `  w ) T y )  <->  ( ( F `
 w ) R ( G `  w
)  ->  ( F `  w ) T ( G `  w ) ) ) )
1410, 13rspc2va 3080 . . . 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 ) ) )
152, 4, 7, 14syl21anc 1217 . . 3  |-  ( (
ph  /\  w  e.  A )  ->  (
( F `  w
) R ( G `
 w )  -> 
( F `  w
) T ( G `
 w ) ) )
1615ralimdva 2794 . 2  |-  ( ph  ->  ( A. w  e.  A  ( F `  w ) R ( G `  w )  ->  A. w  e.  A  ( F `  w ) T ( G `  w ) ) )
17 ffn 5559 . . . 4  |-  ( F : A --> S  ->  F  Fn  A )
181, 17syl 16 . . 3  |-  ( ph  ->  F  Fn  A )
19 ffn 5559 . . . 4  |-  ( G : A --> S  ->  G  Fn  A )
203, 19syl 16 . . 3  |-  ( ph  ->  G  Fn  A )
21 caofref.1 . . 3  |-  ( ph  ->  A  e.  V )
22 inidm 3559 . . 3  |-  ( A  i^i  A )  =  A
23 eqidd 2444 . . 3  |-  ( (
ph  /\  w  e.  A )  ->  ( F `  w )  =  ( F `  w ) )
24 eqidd 2444 . . 3  |-  ( (
ph  /\  w  e.  A )  ->  ( G `  w )  =  ( G `  w ) )
2518, 20, 21, 21, 22, 23, 24ofrfval 6328 . 2  |-  ( ph  ->  ( F  oR R G  <->  A. w  e.  A  ( F `  w ) R ( G `  w ) ) )
2618, 20, 21, 21, 22, 23, 24ofrfval 6328 . 2  |-  ( ph  ->  ( F  oR T G  <->  A. w  e.  A  ( F `  w ) T ( G `  w ) ) )
2716, 25, 263imtr4d 268 1  |-  ( ph  ->  ( F  oR R G  ->  F  oR T G ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2715   class class class wbr 4292    Fn wfn 5413   -->wf 5414   ` cfv 5418    oRcofr 6319
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pr 4531
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-ofr 6321
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator