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Theorem ofrfval 5720
Description: Value of a relation applied to two functions. (Contributed by Mario Carneiro, 28-Jul-2014.)
Hypotheses
Ref Expression
offval.1  |-  ( ph  ->  F  Fn  A )
offval.2  |-  ( ph  ->  G  Fn  B )
offval.3  |-  ( ph  ->  A  e.  V )
offval.4  |-  ( ph  ->  B  e.  W )
offval.5  |-  ( A  i^i  B )  =  S
offval.6  |-  ( (
ph  /\  x  e.  A )  ->  ( F `  x )  =  C )
offval.7  |-  ( (
ph  /\  x  e.  B )  ->  ( G `  x )  =  D )
Assertion
Ref Expression
ofrfval  |-  ( ph  ->  ( F  oR R G  <->  A. x  e.  S  C R D ) )
Distinct variable groups:    x, A    x, F    x, G    ph, x    x, S    x, R
Allowed substitution hints:    B( x)    C( x)    D( x)    V( x)    W( x)

Proof of Theorem ofrfval
Dummy variables  f  g are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 offval.1 . . . 4  |-  ( ph  ->  F  Fn  A )
2 offval.3 . . . 4  |-  ( ph  ->  A  e.  V )
3 fnex 5383 . . . 4  |-  ( ( F  Fn  A  /\  A  e.  V )  ->  F  e.  _V )
41, 2, 3syl2anc 391 . . 3  |-  ( ph  ->  F  e.  _V )
5 offval.2 . . . 4  |-  ( ph  ->  G  Fn  B )
6 offval.4 . . . 4  |-  ( ph  ->  B  e.  W )
7 fnex 5383 . . . 4  |-  ( ( G  Fn  B  /\  B  e.  W )  ->  G  e.  _V )
85, 6, 7syl2anc 391 . . 3  |-  ( ph  ->  G  e.  _V )
9 dmeq 4535 . . . . . 6  |-  ( f  =  F  ->  dom  f  =  dom  F )
10 dmeq 4535 . . . . . 6  |-  ( g  =  G  ->  dom  g  =  dom  G )
119, 10ineqan12d 3140 . . . . 5  |-  ( ( f  =  F  /\  g  =  G )  ->  ( dom  f  i^i 
dom  g )  =  ( dom  F  i^i  dom 
G ) )
12 fveq1 5177 . . . . . 6  |-  ( f  =  F  ->  (
f `  x )  =  ( F `  x ) )
13 fveq1 5177 . . . . . 6  |-  ( g  =  G  ->  (
g `  x )  =  ( G `  x ) )
1412, 13breqan12d 3779 . . . . 5  |-  ( ( f  =  F  /\  g  =  G )  ->  ( ( f `  x ) R ( g `  x )  <-> 
( F `  x
) R ( G `
 x ) ) )
1511, 14raleqbidv 2517 . . . 4  |-  ( ( f  =  F  /\  g  =  G )  ->  ( A. x  e.  ( dom  f  i^i 
dom  g ) ( f `  x ) R ( g `  x )  <->  A. x  e.  ( dom  F  i^i  dom 
G ) ( F `
 x ) R ( G `  x
) ) )
16 df-ofr 5713 . . . 4  |-  oR R  =  { <. f ,  g >.  |  A. x  e.  ( dom  f  i^i  dom  g )
( f `  x
) R ( g `
 x ) }
1715, 16brabga 4001 . . 3  |-  ( ( F  e.  _V  /\  G  e.  _V )  ->  ( F  oR R G  <->  A. x  e.  ( dom  F  i^i  dom 
G ) ( F `
 x ) R ( G `  x
) ) )
184, 8, 17syl2anc 391 . 2  |-  ( ph  ->  ( F  oR R G  <->  A. x  e.  ( dom  F  i^i  dom 
G ) ( F `
 x ) R ( G `  x
) ) )
19 fndm 4998 . . . . . 6  |-  ( F  Fn  A  ->  dom  F  =  A )
201, 19syl 14 . . . . 5  |-  ( ph  ->  dom  F  =  A )
21 fndm 4998 . . . . . 6  |-  ( G  Fn  B  ->  dom  G  =  B )
225, 21syl 14 . . . . 5  |-  ( ph  ->  dom  G  =  B )
2320, 22ineq12d 3139 . . . 4  |-  ( ph  ->  ( dom  F  i^i  dom 
G )  =  ( A  i^i  B ) )
24 offval.5 . . . 4  |-  ( A  i^i  B )  =  S
2523, 24syl6eq 2088 . . 3  |-  ( ph  ->  ( dom  F  i^i  dom 
G )  =  S )
2625raleqdv 2511 . 2  |-  ( ph  ->  ( A. x  e.  ( dom  F  i^i  dom 
G ) ( F `
 x ) R ( G `  x
)  <->  A. x  e.  S  ( F `  x ) R ( G `  x ) ) )
27 inss1 3157 . . . . . . 7  |-  ( A  i^i  B )  C_  A
2824, 27eqsstr3i 2976 . . . . . 6  |-  S  C_  A
2928sseli 2941 . . . . 5  |-  ( x  e.  S  ->  x  e.  A )
30 offval.6 . . . . 5  |-  ( (
ph  /\  x  e.  A )  ->  ( F `  x )  =  C )
3129, 30sylan2 270 . . . 4  |-  ( (
ph  /\  x  e.  S )  ->  ( F `  x )  =  C )
32 inss2 3158 . . . . . . 7  |-  ( A  i^i  B )  C_  B
3324, 32eqsstr3i 2976 . . . . . 6  |-  S  C_  B
3433sseli 2941 . . . . 5  |-  ( x  e.  S  ->  x  e.  B )
35 offval.7 . . . . 5  |-  ( (
ph  /\  x  e.  B )  ->  ( G `  x )  =  D )
3634, 35sylan2 270 . . . 4  |-  ( (
ph  /\  x  e.  S )  ->  ( G `  x )  =  D )
3731, 36breq12d 3777 . . 3  |-  ( (
ph  /\  x  e.  S )  ->  (
( F `  x
) R ( G `
 x )  <->  C R D ) )
3837ralbidva 2322 . 2  |-  ( ph  ->  ( A. x  e.  S  ( F `  x ) R ( G `  x )  <->  A. x  e.  S  C R D ) )
3918, 26, 383bitrd 203 1  |-  ( ph  ->  ( F  oR R G  <->  A. x  e.  S  C R D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 97    <-> wb 98    = wceq 1243    e. wcel 1393   A.wral 2306   _Vcvv 2557    i^i cin 2916   class class class wbr 3764   dom cdm 4345    Fn wfn 4897   ` cfv 4902    oRcofr 5711
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-pow 3927  ax-pr 3944
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ofr 5713
This theorem is referenced by:  ofrval  5722  ofrfval2  5727  caofref  5732  caofrss  5735  caoftrn  5736
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