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Theorem ofrfval 6245
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  o R 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 5893 . . . 4  |-  ( ( F  Fn  A  /\  A  e.  V )  ->  F  e.  _V )
41, 2, 3syl2anc 643 . . 3  |-  ( ph  ->  F  e.  _V )
5 offval.2 . . . 4  |-  ( ph  ->  G  Fn  B )
6 offval.4 . . . 4  |-  ( ph  ->  B  e.  W )
7 fnex 5893 . . . 4  |-  ( ( G  Fn  B  /\  B  e.  W )  ->  G  e.  _V )
85, 6, 7syl2anc 643 . . 3  |-  ( ph  ->  G  e.  _V )
9 dmeq 5003 . . . . . 6  |-  ( f  =  F  ->  dom  f  =  dom  F )
10 dmeq 5003 . . . . . 6  |-  ( g  =  G  ->  dom  g  =  dom  G )
119, 10ineqan12d 3480 . . . . 5  |-  ( ( f  =  F  /\  g  =  G )  ->  ( dom  f  i^i 
dom  g )  =  ( dom  F  i^i  dom 
G ) )
12 fveq1 5660 . . . . . 6  |-  ( f  =  F  ->  (
f `  x )  =  ( F `  x ) )
13 fveq1 5660 . . . . . 6  |-  ( g  =  G  ->  (
g `  x )  =  ( G `  x ) )
1412, 13breqan12d 4161 . . . . 5  |-  ( ( f  =  F  /\  g  =  G )  ->  ( ( f `  x ) R ( g `  x )  <-> 
( F `  x
) R ( G `
 x ) ) )
1511, 14raleqbidv 2852 . . . 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 6238 . . . 4  |-  o R R  =  { <. f ,  g >.  |  A. x  e.  ( dom  f  i^i  dom  g )
( f `  x
) R ( g `
 x ) }
1715, 16brabga 4403 . . 3  |-  ( ( F  e.  _V  /\  G  e.  _V )  ->  ( F  o R R G  <->  A. x  e.  ( dom  F  i^i  dom 
G ) ( F `
 x ) R ( G `  x
) ) )
184, 8, 17syl2anc 643 . 2  |-  ( ph  ->  ( F  o R R G  <->  A. x  e.  ( dom  F  i^i  dom 
G ) ( F `
 x ) R ( G `  x
) ) )
19 fndm 5477 . . . . . 6  |-  ( F  Fn  A  ->  dom  F  =  A )
201, 19syl 16 . . . . 5  |-  ( ph  ->  dom  F  =  A )
21 fndm 5477 . . . . . 6  |-  ( G  Fn  B  ->  dom  G  =  B )
225, 21syl 16 . . . . 5  |-  ( ph  ->  dom  G  =  B )
2320, 22ineq12d 3479 . . . 4  |-  ( ph  ->  ( dom  F  i^i  dom 
G )  =  ( A  i^i  B ) )
24 offval.5 . . . 4  |-  ( A  i^i  B )  =  S
2523, 24syl6eq 2428 . . 3  |-  ( ph  ->  ( dom  F  i^i  dom 
G )  =  S )
2625raleqdv 2846 . 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 3497 . . . . . . 7  |-  ( A  i^i  B )  C_  A
2824, 27eqsstr3i 3315 . . . . . 6  |-  S  C_  A
2928sseli 3280 . . . . 5  |-  ( x  e.  S  ->  x  e.  A )
30 offval.6 . . . . 5  |-  ( (
ph  /\  x  e.  A )  ->  ( F `  x )  =  C )
3129, 30sylan2 461 . . . 4  |-  ( (
ph  /\  x  e.  S )  ->  ( F `  x )  =  C )
32 inss2 3498 . . . . . . 7  |-  ( A  i^i  B )  C_  B
3324, 32eqsstr3i 3315 . . . . . 6  |-  S  C_  B
3433sseli 3280 . . . . 5  |-  ( x  e.  S  ->  x  e.  B )
35 offval.7 . . . . 5  |-  ( (
ph  /\  x  e.  B )  ->  ( G `  x )  =  D )
3634, 35sylan2 461 . . . 4  |-  ( (
ph  /\  x  e.  S )  ->  ( G `  x )  =  D )
3731, 36breq12d 4159 . . 3  |-  ( (
ph  /\  x  e.  S )  ->  (
( F `  x
) R ( G `
 x )  <->  C R D ) )
3837ralbidva 2658 . 2  |-  ( ph  ->  ( A. x  e.  S  ( F `  x ) R ( G `  x )  <->  A. x  e.  S  C R D ) )
3918, 26, 383bitrd 271 1  |-  ( ph  ->  ( F  o R R G  <->  A. x  e.  S  C R D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717   A.wral 2642   _Vcvv 2892    i^i cin 3255   class class class wbr 4146   dom cdm 4811    Fn wfn 5382   ` cfv 5387    o Rcofr 6236
This theorem is referenced by:  ofrval  6247  ofrfval2  6255  caofref  6262  caofrss  6269  caoftrn  6271  ofsubge0  9924  pwsle  13634  pwsleval  13635  psrbaglesupp  16353  psrbagcon  16356  psrbaglefi  16357  psrlidm  16387  0plef  19424  0pledm  19425  itg1ge0  19438  mbfi1fseqlem5  19471  xrge0f  19483  itg2ge0  19487  itg2lea  19496  itg2splitlem  19500  itg2monolem1  19502  itg2mono  19505  itg2i1fseqle  19506  itg2i1fseq  19507  itg2addlem  19510  itg2cnlem1  19513  itg2addnclem  25950  itg2addnc  25952
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
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