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Theorem ofrfval 6543
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 6138 . . . 4  |-  ( ( F  Fn  A  /\  A  e.  V )  ->  F  e.  _V )
41, 2, 3syl2anc 661 . . 3  |-  ( ph  ->  F  e.  _V )
5 offval.2 . . . 4  |-  ( ph  ->  G  Fn  B )
6 offval.4 . . . 4  |-  ( ph  ->  B  e.  W )
7 fnex 6138 . . . 4  |-  ( ( G  Fn  B  /\  B  e.  W )  ->  G  e.  _V )
85, 6, 7syl2anc 661 . . 3  |-  ( ph  ->  G  e.  _V )
9 dmeq 5209 . . . . . 6  |-  ( f  =  F  ->  dom  f  =  dom  F )
10 dmeq 5209 . . . . . 6  |-  ( g  =  G  ->  dom  g  =  dom  G )
119, 10ineqan12d 3707 . . . . 5  |-  ( ( f  =  F  /\  g  =  G )  ->  ( dom  f  i^i 
dom  g )  =  ( dom  F  i^i  dom 
G ) )
12 fveq1 5871 . . . . . 6  |-  ( f  =  F  ->  (
f `  x )  =  ( F `  x ) )
13 fveq1 5871 . . . . . 6  |-  ( g  =  G  ->  (
g `  x )  =  ( G `  x ) )
1412, 13breqan12d 4468 . . . . 5  |-  ( ( f  =  F  /\  g  =  G )  ->  ( ( f `  x ) R ( g `  x )  <-> 
( F `  x
) R ( G `
 x ) ) )
1511, 14raleqbidv 3077 . . . 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 6536 . . . 4  |-  oR R  =  { <. f ,  g >.  |  A. x  e.  ( dom  f  i^i  dom  g )
( f `  x
) R ( g `
 x ) }
1715, 16brabga 4767 . . 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 661 . 2  |-  ( ph  ->  ( F  oR R G  <->  A. x  e.  ( dom  F  i^i  dom 
G ) ( F `
 x ) R ( G `  x
) ) )
19 fndm 5686 . . . . . 6  |-  ( F  Fn  A  ->  dom  F  =  A )
201, 19syl 16 . . . . 5  |-  ( ph  ->  dom  F  =  A )
21 fndm 5686 . . . . . 6  |-  ( G  Fn  B  ->  dom  G  =  B )
225, 21syl 16 . . . . 5  |-  ( ph  ->  dom  G  =  B )
2320, 22ineq12d 3706 . . . 4  |-  ( ph  ->  ( dom  F  i^i  dom 
G )  =  ( A  i^i  B ) )
24 offval.5 . . . 4  |-  ( A  i^i  B )  =  S
2523, 24syl6eq 2524 . . 3  |-  ( ph  ->  ( dom  F  i^i  dom 
G )  =  S )
2625raleqdv 3069 . 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 3723 . . . . . . 7  |-  ( A  i^i  B )  C_  A
2824, 27eqsstr3i 3540 . . . . . 6  |-  S  C_  A
2928sseli 3505 . . . . 5  |-  ( x  e.  S  ->  x  e.  A )
30 offval.6 . . . . 5  |-  ( (
ph  /\  x  e.  A )  ->  ( F `  x )  =  C )
3129, 30sylan2 474 . . . 4  |-  ( (
ph  /\  x  e.  S )  ->  ( F `  x )  =  C )
32 inss2 3724 . . . . . . 7  |-  ( A  i^i  B )  C_  B
3324, 32eqsstr3i 3540 . . . . . 6  |-  S  C_  B
3433sseli 3505 . . . . 5  |-  ( x  e.  S  ->  x  e.  B )
35 offval.7 . . . . 5  |-  ( (
ph  /\  x  e.  B )  ->  ( G `  x )  =  D )
3634, 35sylan2 474 . . . 4  |-  ( (
ph  /\  x  e.  S )  ->  ( G `  x )  =  D )
3731, 36breq12d 4466 . . 3  |-  ( (
ph  /\  x  e.  S )  ->  (
( F `  x
) R ( G `
 x )  <->  C R D ) )
3837ralbidva 2903 . 2  |-  ( ph  ->  ( A. x  e.  S  ( F `  x ) R ( G `  x )  <->  A. x  e.  S  C R D ) )
3918, 26, 383bitrd 279 1  |-  ( ph  ->  ( F  oR R G  <->  A. x  e.  S  C R D ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2817   _Vcvv 3118    i^i cin 3480   class class class wbr 4453   dom cdm 5005    Fn wfn 5589   ` cfv 5594    oRcofr 6534
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pr 4692
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ofr 6536
This theorem is referenced by:  ofrval  6545  ofrfval2  6552  caofref  6561  caofrss  6568  caoftrn  6570  ofsubge0  10547  pwsle  14763  pwsleval  14764  psrbaglesupp  17885  psrbaglesuppOLD  17886  psrbagcon  17890  psrbaglefi  17891  psrbaglefiOLD  17892  psrlidm  17924  psrlidmOLD  17925  0plef  21945  0pledm  21946  itg1ge0  21959  mbfi1fseqlem5  21992  xrge0f  22004  itg2ge0  22008  itg2lea  22017  itg2splitlem  22021  itg2monolem1  22023  itg2mono  22026  itg2i1fseqle  22027  itg2i1fseq  22028  itg2addlem  22031  itg2cnlem1  22034  itg2addnclem  30000
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