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Theorem ofrfval 6566
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 6157 . . . 4  |-  ( ( F  Fn  A  /\  A  e.  V )  ->  F  e.  _V )
41, 2, 3syl2anc 671 . . 3  |-  ( ph  ->  F  e.  _V )
5 offval.2 . . . 4  |-  ( ph  ->  G  Fn  B )
6 offval.4 . . . 4  |-  ( ph  ->  B  e.  W )
7 fnex 6157 . . . 4  |-  ( ( G  Fn  B  /\  B  e.  W )  ->  G  e.  _V )
85, 6, 7syl2anc 671 . . 3  |-  ( ph  ->  G  e.  _V )
9 dmeq 5054 . . . . . 6  |-  ( f  =  F  ->  dom  f  =  dom  F )
10 dmeq 5054 . . . . . 6  |-  ( g  =  G  ->  dom  g  =  dom  G )
119, 10ineqan12d 3648 . . . . 5  |-  ( ( f  =  F  /\  g  =  G )  ->  ( dom  f  i^i 
dom  g )  =  ( dom  F  i^i  dom 
G ) )
12 fveq1 5887 . . . . . 6  |-  ( f  =  F  ->  (
f `  x )  =  ( F `  x ) )
13 fveq1 5887 . . . . . 6  |-  ( g  =  G  ->  (
g `  x )  =  ( G `  x ) )
1412, 13breqan12d 4432 . . . . 5  |-  ( ( f  =  F  /\  g  =  G )  ->  ( ( f `  x ) R ( g `  x )  <-> 
( F `  x
) R ( G `
 x ) ) )
1511, 14raleqbidv 3013 . . . 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 6559 . . . 4  |-  oR R  =  { <. f ,  g >.  |  A. x  e.  ( dom  f  i^i  dom  g )
( f `  x
) R ( g `
 x ) }
1715, 16brabga 4729 . . 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 671 . 2  |-  ( ph  ->  ( F  oR R G  <->  A. x  e.  ( dom  F  i^i  dom 
G ) ( F `
 x ) R ( G `  x
) ) )
19 fndm 5697 . . . . . 6  |-  ( F  Fn  A  ->  dom  F  =  A )
201, 19syl 17 . . . . 5  |-  ( ph  ->  dom  F  =  A )
21 fndm 5697 . . . . . 6  |-  ( G  Fn  B  ->  dom  G  =  B )
225, 21syl 17 . . . . 5  |-  ( ph  ->  dom  G  =  B )
2320, 22ineq12d 3647 . . . 4  |-  ( ph  ->  ( dom  F  i^i  dom 
G )  =  ( A  i^i  B ) )
24 offval.5 . . . 4  |-  ( A  i^i  B )  =  S
2523, 24syl6eq 2512 . . 3  |-  ( ph  ->  ( dom  F  i^i  dom 
G )  =  S )
2625raleqdv 3005 . 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 3664 . . . . . . 7  |-  ( A  i^i  B )  C_  A
2824, 27eqsstr3i 3475 . . . . . 6  |-  S  C_  A
2928sseli 3440 . . . . 5  |-  ( x  e.  S  ->  x  e.  A )
30 offval.6 . . . . 5  |-  ( (
ph  /\  x  e.  A )  ->  ( F `  x )  =  C )
3129, 30sylan2 481 . . . 4  |-  ( (
ph  /\  x  e.  S )  ->  ( F `  x )  =  C )
32 inss2 3665 . . . . . . 7  |-  ( A  i^i  B )  C_  B
3324, 32eqsstr3i 3475 . . . . . 6  |-  S  C_  B
3433sseli 3440 . . . . 5  |-  ( x  e.  S  ->  x  e.  B )
35 offval.7 . . . . 5  |-  ( (
ph  /\  x  e.  B )  ->  ( G `  x )  =  D )
3634, 35sylan2 481 . . . 4  |-  ( (
ph  /\  x  e.  S )  ->  ( G `  x )  =  D )
3731, 36breq12d 4429 . . 3  |-  ( (
ph  /\  x  e.  S )  ->  (
( F `  x
) R ( G `
 x )  <->  C R D ) )
3837ralbidva 2836 . 2  |-  ( ph  ->  ( A. x  e.  S  ( F `  x ) R ( G `  x )  <->  A. x  e.  S  C R D ) )
3918, 26, 383bitrd 287 1  |-  ( ph  ->  ( F  oR R G  <->  A. x  e.  S  C R D ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 375    = wceq 1455    e. wcel 1898   A.wral 2749   _Vcvv 3057    i^i cin 3415   class class class wbr 4416   dom cdm 4853    Fn wfn 5596   ` cfv 5601    oRcofr 6557
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-rep 4529  ax-sep 4539  ax-nul 4548  ax-pr 4653
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-reu 2756  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4213  df-iun 4294  df-br 4417  df-opab 4476  df-mpt 4477  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-ofr 6559
This theorem is referenced by:  ofrval  6568  ofrfval2  6576  caofref  6584  caofrss  6591  caoftrn  6593  ofsubge0  10636  pwsle  15439  pwsleval  15440  psrbaglesupp  18641  psrbagcon  18644  psrbaglefi  18645  psrlidm  18676  0plef  22679  0pledm  22680  itg1ge0  22693  mbfi1fseqlem5  22726  xrge0f  22738  itg2ge0  22742  itg2lea  22751  itg2splitlem  22755  itg2monolem1  22757  itg2mono  22760  itg2i1fseqle  22761  itg2i1fseq  22762  itg2addlem  22765  itg2cnlem1  22768  itg2addnclem  32038
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