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Theorem ofrfval2 6552
Description: The function relation acting on maps. (Contributed by Mario Carneiro, 20-Jul-2014.)
Hypotheses
Ref Expression
offval2.1  |-  ( ph  ->  A  e.  V )
offval2.2  |-  ( (
ph  /\  x  e.  A )  ->  B  e.  W )
offval2.3  |-  ( (
ph  /\  x  e.  A )  ->  C  e.  X )
offval2.4  |-  ( ph  ->  F  =  ( x  e.  A  |->  B ) )
offval2.5  |-  ( ph  ->  G  =  ( x  e.  A  |->  C ) )
Assertion
Ref Expression
ofrfval2  |-  ( ph  ->  ( F  oR R G  <->  A. x  e.  A  B R C ) )
Distinct variable groups:    x, A    ph, x    x, R
Allowed substitution hints:    B( x)    C( x)    F( x)    G( x)    V( x)    W( x)    X( x)

Proof of Theorem ofrfval2
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 offval2.2 . . . . . 6  |-  ( (
ph  /\  x  e.  A )  ->  B  e.  W )
21ralrimiva 2881 . . . . 5  |-  ( ph  ->  A. x  e.  A  B  e.  W )
3 eqid 2467 . . . . . 6  |-  ( x  e.  A  |->  B )  =  ( x  e.  A  |->  B )
43fnmpt 5713 . . . . 5  |-  ( A. x  e.  A  B  e.  W  ->  ( x  e.  A  |->  B )  Fn  A )
52, 4syl 16 . . . 4  |-  ( ph  ->  ( x  e.  A  |->  B )  Fn  A
)
6 offval2.4 . . . . 5  |-  ( ph  ->  F  =  ( x  e.  A  |->  B ) )
76fneq1d 5677 . . . 4  |-  ( ph  ->  ( F  Fn  A  <->  ( x  e.  A  |->  B )  Fn  A ) )
85, 7mpbird 232 . . 3  |-  ( ph  ->  F  Fn  A )
9 offval2.3 . . . . . 6  |-  ( (
ph  /\  x  e.  A )  ->  C  e.  X )
109ralrimiva 2881 . . . . 5  |-  ( ph  ->  A. x  e.  A  C  e.  X )
11 eqid 2467 . . . . . 6  |-  ( x  e.  A  |->  C )  =  ( x  e.  A  |->  C )
1211fnmpt 5713 . . . . 5  |-  ( A. x  e.  A  C  e.  X  ->  ( x  e.  A  |->  C )  Fn  A )
1310, 12syl 16 . . . 4  |-  ( ph  ->  ( x  e.  A  |->  C )  Fn  A
)
14 offval2.5 . . . . 5  |-  ( ph  ->  G  =  ( x  e.  A  |->  C ) )
1514fneq1d 5677 . . . 4  |-  ( ph  ->  ( G  Fn  A  <->  ( x  e.  A  |->  C )  Fn  A ) )
1613, 15mpbird 232 . . 3  |-  ( ph  ->  G  Fn  A )
17 offval2.1 . . 3  |-  ( ph  ->  A  e.  V )
18 inidm 3712 . . 3  |-  ( A  i^i  A )  =  A
196adantr 465 . . . 4  |-  ( (
ph  /\  y  e.  A )  ->  F  =  ( x  e.  A  |->  B ) )
2019fveq1d 5874 . . 3  |-  ( (
ph  /\  y  e.  A )  ->  ( F `  y )  =  ( ( x  e.  A  |->  B ) `
 y ) )
2114adantr 465 . . . 4  |-  ( (
ph  /\  y  e.  A )  ->  G  =  ( x  e.  A  |->  C ) )
2221fveq1d 5874 . . 3  |-  ( (
ph  /\  y  e.  A )  ->  ( G `  y )  =  ( ( x  e.  A  |->  C ) `
 y ) )
238, 16, 17, 17, 18, 20, 22ofrfval 6543 . 2  |-  ( ph  ->  ( F  oR R G  <->  A. y  e.  A  ( (
x  e.  A  |->  B ) `  y ) R ( ( x  e.  A  |->  C ) `
 y ) ) )
24 nffvmpt1 5880 . . . . 5  |-  F/_ x
( ( x  e.  A  |->  B ) `  y )
25 nfcv 2629 . . . . 5  |-  F/_ x R
26 nffvmpt1 5880 . . . . 5  |-  F/_ x
( ( x  e.  A  |->  C ) `  y )
2724, 25, 26nfbr 4497 . . . 4  |-  F/ x
( ( x  e.  A  |->  B ) `  y ) R ( ( x  e.  A  |->  C ) `  y
)
28 nfv 1683 . . . 4  |-  F/ y ( ( x  e.  A  |->  B ) `  x ) R ( ( x  e.  A  |->  C ) `  x
)
29 fveq2 5872 . . . . 5  |-  ( y  =  x  ->  (
( x  e.  A  |->  B ) `  y
)  =  ( ( x  e.  A  |->  B ) `  x ) )
30 fveq2 5872 . . . . 5  |-  ( y  =  x  ->  (
( x  e.  A  |->  C ) `  y
)  =  ( ( x  e.  A  |->  C ) `  x ) )
3129, 30breq12d 4466 . . . 4  |-  ( y  =  x  ->  (
( ( x  e.  A  |->  B ) `  y ) R ( ( x  e.  A  |->  C ) `  y
)  <->  ( ( x  e.  A  |->  B ) `
 x ) R ( ( x  e.  A  |->  C ) `  x ) ) )
3227, 28, 31cbvral 3089 . . 3  |-  ( A. y  e.  A  (
( x  e.  A  |->  B ) `  y
) R ( ( x  e.  A  |->  C ) `  y )  <->  A. x  e.  A  ( ( x  e.  A  |->  B ) `  x ) R ( ( x  e.  A  |->  C ) `  x
) )
33 simpr 461 . . . . . 6  |-  ( (
ph  /\  x  e.  A )  ->  x  e.  A )
343fvmpt2 5964 . . . . . 6  |-  ( ( x  e.  A  /\  B  e.  W )  ->  ( ( x  e.  A  |->  B ) `  x )  =  B )
3533, 1, 34syl2anc 661 . . . . 5  |-  ( (
ph  /\  x  e.  A )  ->  (
( x  e.  A  |->  B ) `  x
)  =  B )
3611fvmpt2 5964 . . . . . 6  |-  ( ( x  e.  A  /\  C  e.  X )  ->  ( ( x  e.  A  |->  C ) `  x )  =  C )
3733, 9, 36syl2anc 661 . . . . 5  |-  ( (
ph  /\  x  e.  A )  ->  (
( x  e.  A  |->  C ) `  x
)  =  C )
3835, 37breq12d 4466 . . . 4  |-  ( (
ph  /\  x  e.  A )  ->  (
( ( x  e.  A  |->  B ) `  x ) R ( ( x  e.  A  |->  C ) `  x
)  <->  B R C ) )
3938ralbidva 2903 . . 3  |-  ( ph  ->  ( A. x  e.  A  ( ( x  e.  A  |->  B ) `
 x ) R ( ( x  e.  A  |->  C ) `  x )  <->  A. x  e.  A  B R C ) )
4032, 39syl5bb 257 . 2  |-  ( ph  ->  ( A. y  e.  A  ( ( x  e.  A  |->  B ) `
 y ) R ( ( x  e.  A  |->  C ) `  y )  <->  A. x  e.  A  B R C ) )
4123, 40bitrd 253 1  |-  ( ph  ->  ( F  oR R G  <->  A. x  e.  A  B R C ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2817   class class class wbr 4453    |-> cmpt 4511    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-8 1769  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-pow 4631  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:  gsumbagdiaglem  17897  mplmonmul  17996  coe1mul2lem1  18178  itg2const  22015  itg2const2  22016  itg2uba  22018  itg2mulclem  22021  itg2splitlem  22023  itg2split  22024  itg2monolem1  22025  itg2gt0  22035  itg2cnlem1  22036  itg2cnlem2  22037  iblss  22079  i1fibl  22082  itgitg1  22083  itgle  22084  ibladdlem  22094  iblabs  22103  iblabsr  22104  iblmulc2  22105  bddmulibl  22113  itg2addnclem  29993  itg2addnclem3  29995  itg2addnc  29996  itg2gt0cn  29997  ibladdnclem  29998  iblabsnc  30006  iblmulc2nc  30007  bddiblnc  30012  ftc1anclem4  30020  ftc1anclem5  30021  ftc1anclem6  30022  ftc1anclem7  30023  ftc1anclem8  30024  ftc1anc  30025
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