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Theorem ofrfval2 6096
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  o R 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 2626 . . . . 5  |-  ( ph  ->  A. x  e.  A  B  e.  W )
3 eqid 2283 . . . . . 6  |-  ( x  e.  A  |->  B )  =  ( x  e.  A  |->  B )
43fnmpt 5370 . . . . 5  |-  ( A. x  e.  A  B  e.  W  ->  ( x  e.  A  |->  B )  Fn  A )
52, 4syl 15 . . . 4  |-  ( ph  ->  ( x  e.  A  |->  B )  Fn  A
)
6 offval2.4 . . . . 5  |-  ( ph  ->  F  =  ( x  e.  A  |->  B ) )
76fneq1d 5335 . . . 4  |-  ( ph  ->  ( F  Fn  A  <->  ( x  e.  A  |->  B )  Fn  A ) )
85, 7mpbird 223 . . 3  |-  ( ph  ->  F  Fn  A )
9 offval2.3 . . . . . 6  |-  ( (
ph  /\  x  e.  A )  ->  C  e.  X )
109ralrimiva 2626 . . . . 5  |-  ( ph  ->  A. x  e.  A  C  e.  X )
11 eqid 2283 . . . . . 6  |-  ( x  e.  A  |->  C )  =  ( x  e.  A  |->  C )
1211fnmpt 5370 . . . . 5  |-  ( A. x  e.  A  C  e.  X  ->  ( x  e.  A  |->  C )  Fn  A )
1310, 12syl 15 . . . 4  |-  ( ph  ->  ( x  e.  A  |->  C )  Fn  A
)
14 offval2.5 . . . . 5  |-  ( ph  ->  G  =  ( x  e.  A  |->  C ) )
1514fneq1d 5335 . . . 4  |-  ( ph  ->  ( G  Fn  A  <->  ( x  e.  A  |->  C )  Fn  A ) )
1613, 15mpbird 223 . . 3  |-  ( ph  ->  G  Fn  A )
17 offval2.1 . . 3  |-  ( ph  ->  A  e.  V )
18 inidm 3378 . . 3  |-  ( A  i^i  A )  =  A
196adantr 451 . . . 4  |-  ( (
ph  /\  y  e.  A )  ->  F  =  ( x  e.  A  |->  B ) )
2019fveq1d 5527 . . 3  |-  ( (
ph  /\  y  e.  A )  ->  ( F `  y )  =  ( ( x  e.  A  |->  B ) `
 y ) )
2114adantr 451 . . . 4  |-  ( (
ph  /\  y  e.  A )  ->  G  =  ( x  e.  A  |->  C ) )
2221fveq1d 5527 . . 3  |-  ( (
ph  /\  y  e.  A )  ->  ( G `  y )  =  ( ( x  e.  A  |->  C ) `
 y ) )
238, 16, 17, 17, 18, 20, 22ofrfval 6086 . 2  |-  ( ph  ->  ( F  o R R G  <->  A. y  e.  A  ( (
x  e.  A  |->  B ) `  y ) R ( ( x  e.  A  |->  C ) `
 y ) ) )
24 nfmpt1 4109 . . . . . 6  |-  F/_ x
( x  e.  A  |->  B )
25 nfcv 2419 . . . . . 6  |-  F/_ x
y
2624, 25nffv 5532 . . . . 5  |-  F/_ x
( ( x  e.  A  |->  B ) `  y )
27 nfcv 2419 . . . . 5  |-  F/_ x R
28 nfmpt1 4109 . . . . . 6  |-  F/_ x
( x  e.  A  |->  C )
2928, 25nffv 5532 . . . . 5  |-  F/_ x
( ( x  e.  A  |->  C ) `  y )
3026, 27, 29nfbr 4067 . . . 4  |-  F/ x
( ( x  e.  A  |->  B ) `  y ) R ( ( x  e.  A  |->  C ) `  y
)
31 nfv 1605 . . . 4  |-  F/ y ( ( x  e.  A  |->  B ) `  x ) R ( ( x  e.  A  |->  C ) `  x
)
32 fveq2 5525 . . . . 5  |-  ( y  =  x  ->  (
( x  e.  A  |->  B ) `  y
)  =  ( ( x  e.  A  |->  B ) `  x ) )
33 fveq2 5525 . . . . 5  |-  ( y  =  x  ->  (
( x  e.  A  |->  C ) `  y
)  =  ( ( x  e.  A  |->  C ) `  x ) )
3432, 33breq12d 4036 . . . 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 ) ) )
3530, 31, 34cbvral 2760 . . 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
) )
36 simpr 447 . . . . . 6  |-  ( (
ph  /\  x  e.  A )  ->  x  e.  A )
373fvmpt2 5608 . . . . . 6  |-  ( ( x  e.  A  /\  B  e.  W )  ->  ( ( x  e.  A  |->  B ) `  x )  =  B )
3836, 1, 37syl2anc 642 . . . . 5  |-  ( (
ph  /\  x  e.  A )  ->  (
( x  e.  A  |->  B ) `  x
)  =  B )
3911fvmpt2 5608 . . . . . 6  |-  ( ( x  e.  A  /\  C  e.  X )  ->  ( ( x  e.  A  |->  C ) `  x )  =  C )
4036, 9, 39syl2anc 642 . . . . 5  |-  ( (
ph  /\  x  e.  A )  ->  (
( x  e.  A  |->  C ) `  x
)  =  C )
4138, 40breq12d 4036 . . . 4  |-  ( (
ph  /\  x  e.  A )  ->  (
( ( x  e.  A  |->  B ) `  x ) R ( ( x  e.  A  |->  C ) `  x
)  <->  B R C ) )
4241ralbidva 2559 . . 3  |-  ( ph  ->  ( A. x  e.  A  ( ( x  e.  A  |->  B ) `
 x ) R ( ( x  e.  A  |->  C ) `  x )  <->  A. x  e.  A  B R C ) )
4335, 42syl5bb 248 . 2  |-  ( ph  ->  ( A. y  e.  A  ( ( x  e.  A  |->  B ) `
 y ) R ( ( x  e.  A  |->  C ) `  y )  <->  A. x  e.  A  B R C ) )
4423, 43bitrd 244 1  |-  ( ph  ->  ( F  o R R G  <->  A. x  e.  A  B R C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   class class class wbr 4023    e. cmpt 4077    Fn wfn 5250   ` cfv 5255    o Rcofr 6077
This theorem is referenced by:  gsumbagdiaglem  16121  mplmonmul  16208  coe1mul2lem1  16344  itg2const  19095  itg2const2  19096  itg2uba  19098  itg2mulclem  19101  itg2splitlem  19103  itg2split  19104  itg2monolem1  19105  itg2gt0  19115  itg2cnlem1  19116  itg2cnlem2  19117  iblss  19159  i1fibl  19162  itgitg1  19163  itgle  19164  ibladdlem  19174  iblabs  19183  iblabsr  19184  iblmulc2  19185  bddmulibl  19193
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ofr 6079
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