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Theorem offval2 6097
Description: The function operation expressed as a mapping. (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
offval2  |-  ( ph  ->  ( F  oF R G )  =  ( 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 offval2
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 offval2.2 . . . . . 6  |-  ( (
ph  /\  x  e.  A )  ->  B  e.  W )
21ralrimiva 2550 . . . . 5  |-  ( ph  ->  A. x  e.  A  B  e.  W )
3 eqid 2177 . . . . . 6  |-  ( x  e.  A  |->  B )  =  ( x  e.  A  |->  B )
43fnmpt 5342 . . . . 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 5306 . . . 4  |-  ( ph  ->  ( F  Fn  A  <->  ( x  e.  A  |->  B )  Fn  A ) )
85, 7mpbird 167 . . 3  |-  ( ph  ->  F  Fn  A )
9 offval2.3 . . . . . 6  |-  ( (
ph  /\  x  e.  A )  ->  C  e.  X )
109ralrimiva 2550 . . . . 5  |-  ( ph  ->  A. x  e.  A  C  e.  X )
11 eqid 2177 . . . . . 6  |-  ( x  e.  A  |->  C )  =  ( x  e.  A  |->  C )
1211fnmpt 5342 . . . . 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 5306 . . . 4  |-  ( ph  ->  ( G  Fn  A  <->  ( x  e.  A  |->  C )  Fn  A ) )
1613, 15mpbird 167 . . 3  |-  ( ph  ->  G  Fn  A )
17 offval2.1 . . 3  |-  ( ph  ->  A  e.  V )
18 inidm 3344 . . 3  |-  ( A  i^i  A )  =  A
196adantr 276 . . . 4  |-  ( (
ph  /\  y  e.  A )  ->  F  =  ( x  e.  A  |->  B ) )
2019fveq1d 5517 . . 3  |-  ( (
ph  /\  y  e.  A )  ->  ( F `  y )  =  ( ( x  e.  A  |->  B ) `
 y ) )
2114adantr 276 . . . 4  |-  ( (
ph  /\  y  e.  A )  ->  G  =  ( x  e.  A  |->  C ) )
2221fveq1d 5517 . . 3  |-  ( (
ph  /\  y  e.  A )  ->  ( G `  y )  =  ( ( x  e.  A  |->  C ) `
 y ) )
238, 16, 17, 17, 18, 20, 22offval 6089 . 2  |-  ( ph  ->  ( F  oF R G )  =  ( y  e.  A  |->  ( ( ( x  e.  A  |->  B ) `
 y ) R ( ( x  e.  A  |->  C ) `  y ) ) ) )
24 nffvmpt1 5526 . . . . 5  |-  F/_ x
( ( x  e.  A  |->  B ) `  y )
25 nfcv 2319 . . . . 5  |-  F/_ x R
26 nffvmpt1 5526 . . . . 5  |-  F/_ x
( ( x  e.  A  |->  C ) `  y )
2724, 25, 26nfov 5904 . . . 4  |-  F/_ x
( ( ( x  e.  A  |->  B ) `
 y ) R ( ( x  e.  A  |->  C ) `  y ) )
28 nfcv 2319 . . . 4  |-  F/_ y
( ( ( x  e.  A  |->  B ) `
 x ) R ( ( x  e.  A  |->  C ) `  x ) )
29 fveq2 5515 . . . . 5  |-  ( y  =  x  ->  (
( x  e.  A  |->  B ) `  y
)  =  ( ( x  e.  A  |->  B ) `  x ) )
30 fveq2 5515 . . . . 5  |-  ( y  =  x  ->  (
( x  e.  A  |->  C ) `  y
)  =  ( ( x  e.  A  |->  C ) `  x ) )
3129, 30oveq12d 5892 . . . 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, 31cbvmpt 4098 . . 3  |-  ( y  e.  A  |->  ( ( ( x  e.  A  |->  B ) `  y
) R ( ( x  e.  A  |->  C ) `  y ) ) )  =  ( x  e.  A  |->  ( ( ( x  e.  A  |->  B ) `  x ) R ( ( x  e.  A  |->  C ) `  x
) ) )
33 simpr 110 . . . . . 6  |-  ( (
ph  /\  x  e.  A )  ->  x  e.  A )
343fvmpt2 5599 . . . . . 6  |-  ( ( x  e.  A  /\  B  e.  W )  ->  ( ( x  e.  A  |->  B ) `  x )  =  B )
3533, 1, 34syl2anc 411 . . . . 5  |-  ( (
ph  /\  x  e.  A )  ->  (
( x  e.  A  |->  B ) `  x
)  =  B )
3611fvmpt2 5599 . . . . . 6  |-  ( ( x  e.  A  /\  C  e.  X )  ->  ( ( x  e.  A  |->  C ) `  x )  =  C )
3733, 9, 36syl2anc 411 . . . . 5  |-  ( (
ph  /\  x  e.  A )  ->  (
( x  e.  A  |->  C ) `  x
)  =  C )
3835, 37oveq12d 5892 . . . 4  |-  ( (
ph  /\  x  e.  A )  ->  (
( ( x  e.  A  |->  B ) `  x ) R ( ( x  e.  A  |->  C ) `  x
) )  =  ( B R C ) )
3938mpteq2dva 4093 . . 3  |-  ( ph  ->  ( x  e.  A  |->  ( ( ( x  e.  A  |->  B ) `
 x ) R ( ( x  e.  A  |->  C ) `  x ) ) )  =  ( x  e.  A  |->  ( B R C ) ) )
4032, 39eqtrid 2222 . 2  |-  ( ph  ->  ( y  e.  A  |->  ( ( ( x  e.  A  |->  B ) `
 y ) R ( ( x  e.  A  |->  C ) `  y ) ) )  =  ( x  e.  A  |->  ( B R C ) ) )
4123, 40eqtrd 2210 1  |-  ( ph  ->  ( F  oF R G )  =  ( x  e.  A  |->  ( B R C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1353    e. wcel 2148   A.wral 2455    |-> cmpt 4064    Fn wfn 5211   ` cfv 5216  (class class class)co 5874    oFcof 6080
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-coll 4118  ax-sep 4121  ax-pow 4174  ax-pr 4209  ax-setind 4536
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-ov 5877  df-oprab 5878  df-mpo 5879  df-of 6082
This theorem is referenced by:  ofc12  6102  caofinvl  6104  caofcom  6105  dvimulf  14063  dvexp  14068  dvmptaddx  14074  dvmptmulx  14075  dvef  14081
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