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

Theorem caofrss 6269
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 )
21ffvelrnda 5802 . . . 4  |-  ( (
ph  /\  w  e.  A )  ->  ( F `  w )  e.  S )
3 caofcom.3 . . . . 5  |-  ( ph  ->  G : A --> S )
43ffvelrnda 5802 . . . 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 2734 . . . . 5  |-  ( ph  ->  A. x  e.  S  A. y  e.  S  ( x R y  ->  x T y ) )
76adantr 452 . . . 4  |-  ( (
ph  /\  w  e.  A )  ->  A. x  e.  S  A. y  e.  S  ( x R y  ->  x T y ) )
8 breq1 4149 . . . . . 6  |-  ( x  =  ( F `  w )  ->  (
x R y  <->  ( F `  w ) R y ) )
9 breq1 4149 . . . . . 6  |-  ( x  =  ( F `  w )  ->  (
x T y  <->  ( F `  w ) T y ) )
108, 9imbi12d 312 . . . . 5  |-  ( x  =  ( F `  w )  ->  (
( x R y  ->  x T y )  <->  ( ( F `
 w ) R y  ->  ( F `  w ) T y ) ) )
11 breq2 4150 . . . . . 6  |-  ( y  =  ( G `  w )  ->  (
( F `  w
) R y  <->  ( F `  w ) R ( G `  w ) ) )
12 breq2 4150 . . . . . 6  |-  ( y  =  ( G `  w )  ->  (
( F `  w
) T y  <->  ( F `  w ) T ( G `  w ) ) )
1311, 12imbi12d 312 . . . . 5  |-  ( y  =  ( G `  w )  ->  (
( ( F `  w ) R y  ->  ( F `  w ) T y )  <->  ( ( F `
 w ) R ( G `  w
)  ->  ( F `  w ) T ( G `  w ) ) ) )
1410, 13rspc2va 2995 . . . 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 1183 . . 3  |-  ( (
ph  /\  w  e.  A )  ->  (
( F `  w
) R ( G `
 w )  -> 
( F `  w
) T ( G `
 w ) ) )
1615ralimdva 2720 . 2  |-  ( ph  ->  ( A. w  e.  A  ( F `  w ) R ( G `  w )  ->  A. w  e.  A  ( F `  w ) T ( G `  w ) ) )
17 ffn 5524 . . . 4  |-  ( F : A --> S  ->  F  Fn  A )
181, 17syl 16 . . 3  |-  ( ph  ->  F  Fn  A )
19 ffn 5524 . . . 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 3486 . . 3  |-  ( A  i^i  A )  =  A
23 eqidd 2381 . . 3  |-  ( (
ph  /\  w  e.  A )  ->  ( F `  w )  =  ( F `  w ) )
24 eqidd 2381 . . 3  |-  ( (
ph  /\  w  e.  A )  ->  ( G `  w )  =  ( G `  w ) )
2518, 20, 21, 21, 22, 23, 24ofrfval 6245 . 2  |-  ( ph  ->  ( F  o R R G  <->  A. w  e.  A  ( F `  w ) R ( G `  w ) ) )
2618, 20, 21, 21, 22, 23, 24ofrfval 6245 . 2  |-  ( ph  ->  ( F  o R T G  <->  A. w  e.  A  ( F `  w ) T ( G `  w ) ) )
2716, 25, 263imtr4d 260 1  |-  ( ph  ->  ( F  o R R G  ->  F  o R T G ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   A.wral 2642   class class class wbr 4146    Fn wfn 5382   -->wf 5383   ` cfv 5387    o Rcofr 6236
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pr 4337
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-reu 2649  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ofr 6238
  Copyright terms: Public domain W3C validator