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Theorem ofrfval2 5885
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 2447 . . . . 5  |-  ( ph  ->  A. x  e.  A  B  e.  W )
3 eqid 2089 . . . . . 6  |-  ( x  e.  A  |->  B )  =  ( x  e.  A  |->  B )
43fnmpt 5153 . . . . 5  |-  ( A. x  e.  A  B  e.  W  ->  ( x  e.  A  |->  B )  Fn  A )
52, 4syl 14 . . . 4  |-  ( ph  ->  ( x  e.  A  |->  B )  Fn  A
)
6 offval2.4 . . . . 5  |-  ( ph  ->  F  =  ( x  e.  A  |->  B ) )
76fneq1d 5117 . . . 4  |-  ( ph  ->  ( F  Fn  A  <->  ( x  e.  A  |->  B )  Fn  A ) )
85, 7mpbird 166 . . 3  |-  ( ph  ->  F  Fn  A )
9 offval2.3 . . . . . 6  |-  ( (
ph  /\  x  e.  A )  ->  C  e.  X )
109ralrimiva 2447 . . . . 5  |-  ( ph  ->  A. x  e.  A  C  e.  X )
11 eqid 2089 . . . . . 6  |-  ( x  e.  A  |->  C )  =  ( x  e.  A  |->  C )
1211fnmpt 5153 . . . . 5  |-  ( A. x  e.  A  C  e.  X  ->  ( x  e.  A  |->  C )  Fn  A )
1310, 12syl 14 . . . 4  |-  ( ph  ->  ( x  e.  A  |->  C )  Fn  A
)
14 offval2.5 . . . . 5  |-  ( ph  ->  G  =  ( x  e.  A  |->  C ) )
1514fneq1d 5117 . . . 4  |-  ( ph  ->  ( G  Fn  A  <->  ( x  e.  A  |->  C )  Fn  A ) )
1613, 15mpbird 166 . . 3  |-  ( ph  ->  G  Fn  A )
17 offval2.1 . . 3  |-  ( ph  ->  A  e.  V )
18 inidm 3210 . . 3  |-  ( A  i^i  A )  =  A
196adantr 271 . . . 4  |-  ( (
ph  /\  y  e.  A )  ->  F  =  ( x  e.  A  |->  B ) )
2019fveq1d 5320 . . 3  |-  ( (
ph  /\  y  e.  A )  ->  ( F `  y )  =  ( ( x  e.  A  |->  B ) `
 y ) )
2114adantr 271 . . . 4  |-  ( (
ph  /\  y  e.  A )  ->  G  =  ( x  e.  A  |->  C ) )
2221fveq1d 5320 . . 3  |-  ( (
ph  /\  y  e.  A )  ->  ( G `  y )  =  ( ( x  e.  A  |->  C ) `
 y ) )
238, 16, 17, 17, 18, 20, 22ofrfval 5878 . 2  |-  ( ph  ->  ( F  oR R G  <->  A. y  e.  A  ( (
x  e.  A  |->  B ) `  y ) R ( ( x  e.  A  |->  C ) `
 y ) ) )
24 nffvmpt1 5329 . . . . 5  |-  F/_ x
( ( x  e.  A  |->  B ) `  y )
25 nfcv 2229 . . . . 5  |-  F/_ x R
26 nffvmpt1 5329 . . . . 5  |-  F/_ x
( ( x  e.  A  |->  C ) `  y )
2724, 25, 26nfbr 3895 . . . 4  |-  F/ x
( ( x  e.  A  |->  B ) `  y ) R ( ( x  e.  A  |->  C ) `  y
)
28 nfv 1467 . . . 4  |-  F/ y ( ( x  e.  A  |->  B ) `  x ) R ( ( x  e.  A  |->  C ) `  x
)
29 fveq2 5318 . . . . 5  |-  ( y  =  x  ->  (
( x  e.  A  |->  B ) `  y
)  =  ( ( x  e.  A  |->  B ) `  x ) )
30 fveq2 5318 . . . . 5  |-  ( y  =  x  ->  (
( x  e.  A  |->  C ) `  y
)  =  ( ( x  e.  A  |->  C ) `  x ) )
3129, 30breq12d 3864 . . . 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 2587 . . 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 109 . . . . . 6  |-  ( (
ph  /\  x  e.  A )  ->  x  e.  A )
343fvmpt2 5399 . . . . . 6  |-  ( ( x  e.  A  /\  B  e.  W )  ->  ( ( x  e.  A  |->  B ) `  x )  =  B )
3533, 1, 34syl2anc 404 . . . . 5  |-  ( (
ph  /\  x  e.  A )  ->  (
( x  e.  A  |->  B ) `  x
)  =  B )
3611fvmpt2 5399 . . . . . 6  |-  ( ( x  e.  A  /\  C  e.  X )  ->  ( ( x  e.  A  |->  C ) `  x )  =  C )
3733, 9, 36syl2anc 404 . . . . 5  |-  ( (
ph  /\  x  e.  A )  ->  (
( x  e.  A  |->  C ) `  x
)  =  C )
3835, 37breq12d 3864 . . . 4  |-  ( (
ph  /\  x  e.  A )  ->  (
( ( x  e.  A  |->  B ) `  x ) R ( ( x  e.  A  |->  C ) `  x
)  <->  B R C ) )
3938ralbidva 2377 . . 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 191 . 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 187 1  |-  ( ph  ->  ( F  oR R G  <->  A. x  e.  A  B R C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1290    e. wcel 1439   A.wral 2360   class class class wbr 3851    |-> cmpt 3905    Fn wfn 5023   ` cfv 5028    oRcofr 5869
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-coll 3960  ax-sep 3963  ax-pow 4015  ax-pr 4045
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-reu 2367  df-rab 2369  df-v 2622  df-sbc 2842  df-csb 2935  df-un 3004  df-in 3006  df-ss 3013  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-iun 3738  df-br 3852  df-opab 3906  df-mpt 3907  df-id 4129  df-xp 4458  df-rel 4459  df-cnv 4460  df-co 4461  df-dm 4462  df-rn 4463  df-res 4464  df-ima 4465  df-iota 4993  df-fun 5030  df-fn 5031  df-f 5032  df-f1 5033  df-fo 5034  df-f1o 5035  df-fv 5036  df-ofr 5871
This theorem is referenced by: (None)
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