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Theorem ofrfval 6547
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 6140 . . . 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 6140 . . . 4  |-  ( ( G  Fn  B  /\  B  e.  W )  ->  G  e.  _V )
85, 6, 7syl2anc 661 . . 3  |-  ( ph  ->  G  e.  _V )
9 dmeq 5213 . . . . . 6  |-  ( f  =  F  ->  dom  f  =  dom  F )
10 dmeq 5213 . . . . . 6  |-  ( g  =  G  ->  dom  g  =  dom  G )
119, 10ineqan12d 3698 . . . . 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 4471 . . . . 5  |-  ( ( f  =  F  /\  g  =  G )  ->  ( ( f `  x ) R ( g `  x )  <-> 
( F `  x
) R ( G `
 x ) ) )
1511, 14raleqbidv 3068 . . . 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 6540 . . . 4  |-  oR R  =  { <. f ,  g >.  |  A. x  e.  ( dom  f  i^i  dom  g )
( f `  x
) R ( g `
 x ) }
1715, 16brabga 4770 . . 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 3697 . . . 4  |-  ( ph  ->  ( dom  F  i^i  dom 
G )  =  ( A  i^i  B ) )
24 offval.5 . . . 4  |-  ( A  i^i  B )  =  S
2523, 24syl6eq 2514 . . 3  |-  ( ph  ->  ( dom  F  i^i  dom 
G )  =  S )
2625raleqdv 3060 . 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 3714 . . . . . . 7  |-  ( A  i^i  B )  C_  A
2824, 27eqsstr3i 3530 . . . . . 6  |-  S  C_  A
2928sseli 3495 . . . . 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 3715 . . . . . . 7  |-  ( A  i^i  B )  C_  B
3324, 32eqsstr3i 3530 . . . . . 6  |-  S  C_  B
3433sseli 3495 . . . . 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 4469 . . 3  |-  ( (
ph  /\  x  e.  S )  ->  (
( F `  x
) R ( G `
 x )  <->  C R D ) )
3837ralbidva 2893 . 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 1395    e. wcel 1819   A.wral 2807   _Vcvv 3109    i^i cin 3470   class class class wbr 4456   dom cdm 5008    Fn wfn 5589   ` cfv 5594    oRcofr 6538
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  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 6540
This theorem is referenced by:  ofrval  6549  ofrfval2  6556  caofref  6565  caofrss  6572  caoftrn  6574  ofsubge0  10555  pwsle  14909  pwsleval  14910  psrbaglesupp  18144  psrbaglesuppOLD  18145  psrbagcon  18149  psrbaglefi  18150  psrbaglefiOLD  18151  psrlidm  18183  psrlidmOLD  18184  0plef  22205  0pledm  22206  itg1ge0  22219  mbfi1fseqlem5  22252  xrge0f  22264  itg2ge0  22268  itg2lea  22277  itg2splitlem  22281  itg2monolem1  22283  itg2mono  22286  itg2i1fseqle  22287  itg2i1fseq  22288  itg2addlem  22291  itg2cnlem1  22294  itg2addnclem  30250
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