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Theorem offval2f 24111
Description: The function operation expressed as a mapping. (Contributed by Thierry Arnoux, 23-Jun-2017.)
Hypotheses
Ref Expression
offval2f.0  |-  F/ x ph
offval2f.a  |-  F/_ x A
offval2f.1  |-  ( ph  ->  A  e.  V )
offval2f.2  |-  ( (
ph  /\  x  e.  A )  ->  B  e.  W )
offval2f.3  |-  ( (
ph  /\  x  e.  A )  ->  C  e.  X )
offval2f.4  |-  ( ph  ->  F  =  ( x  e.  A  |->  B ) )
offval2f.5  |-  ( ph  ->  G  =  ( x  e.  A  |->  C ) )
Assertion
Ref Expression
offval2f  |-  ( ph  ->  ( F  o F R G )  =  ( x  e.  A  |->  ( B R C ) ) )
Distinct variable group:    x, R
Allowed substitution hints:    ph( x)    A( x)    B( x)    C( x)    F( x)    G( x)    V( x)    W( x)    X( x)

Proof of Theorem offval2f
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 offval2f.0 . . . . . 6  |-  F/ x ph
2 offval2f.2 . . . . . . 7  |-  ( (
ph  /\  x  e.  A )  ->  B  e.  W )
32ex 425 . . . . . 6  |-  ( ph  ->  ( x  e.  A  ->  B  e.  W ) )
41, 3ralrimi 2793 . . . . 5  |-  ( ph  ->  A. x  e.  A  B  e.  W )
5 offval2f.a . . . . . 6  |-  F/_ x A
65fnmptf 24105 . . . . 5  |-  ( A. x  e.  A  B  e.  W  ->  ( x  e.  A  |->  B )  Fn  A )
74, 6syl 16 . . . 4  |-  ( ph  ->  ( x  e.  A  |->  B )  Fn  A
)
8 offval2f.4 . . . . 5  |-  ( ph  ->  F  =  ( x  e.  A  |->  B ) )
98fneq1d 5565 . . . 4  |-  ( ph  ->  ( F  Fn  A  <->  ( x  e.  A  |->  B )  Fn  A ) )
107, 9mpbird 225 . . 3  |-  ( ph  ->  F  Fn  A )
11 offval2f.3 . . . . . . 7  |-  ( (
ph  /\  x  e.  A )  ->  C  e.  X )
1211ex 425 . . . . . 6  |-  ( ph  ->  ( x  e.  A  ->  C  e.  X ) )
131, 12ralrimi 2793 . . . . 5  |-  ( ph  ->  A. x  e.  A  C  e.  X )
145fnmptf 24105 . . . . 5  |-  ( A. x  e.  A  C  e.  X  ->  ( x  e.  A  |->  C )  Fn  A )
1513, 14syl 16 . . . 4  |-  ( ph  ->  ( x  e.  A  |->  C )  Fn  A
)
16 offval2f.5 . . . . 5  |-  ( ph  ->  G  =  ( x  e.  A  |->  C ) )
1716fneq1d 5565 . . . 4  |-  ( ph  ->  ( G  Fn  A  <->  ( x  e.  A  |->  C )  Fn  A ) )
1815, 17mpbird 225 . . 3  |-  ( ph  ->  G  Fn  A )
19 offval2f.1 . . 3  |-  ( ph  ->  A  e.  V )
20 inidm 3535 . . 3  |-  ( A  i^i  A )  =  A
218adantr 453 . . . 4  |-  ( (
ph  /\  y  e.  A )  ->  F  =  ( x  e.  A  |->  B ) )
2221fveq1d 5759 . . 3  |-  ( (
ph  /\  y  e.  A )  ->  ( F `  y )  =  ( ( x  e.  A  |->  B ) `
 y ) )
2316adantr 453 . . . 4  |-  ( (
ph  /\  y  e.  A )  ->  G  =  ( x  e.  A  |->  C ) )
2423fveq1d 5759 . . 3  |-  ( (
ph  /\  y  e.  A )  ->  ( G `  y )  =  ( ( x  e.  A  |->  C ) `
 y ) )
2510, 18, 19, 19, 20, 22, 24offval 6341 . 2  |-  ( ph  ->  ( F  o F R G )  =  ( y  e.  A  |->  ( ( ( x  e.  A  |->  B ) `
 y ) R ( ( x  e.  A  |->  C ) `  y ) ) ) )
26 nfcv 2578 . . . 4  |-  F/_ y A
27 nffvmpt1 5765 . . . . 5  |-  F/_ x
( ( x  e.  A  |->  B ) `  y )
28 nfcv 2578 . . . . 5  |-  F/_ x R
29 nffvmpt1 5765 . . . . 5  |-  F/_ x
( ( x  e.  A  |->  C ) `  y )
3027, 28, 29nfov 6133 . . . 4  |-  F/_ x
( ( ( x  e.  A  |->  B ) `
 y ) R ( ( x  e.  A  |->  C ) `  y ) )
31 nfcv 2578 . . . 4  |-  F/_ y
( ( ( x  e.  A  |->  B ) `
 x ) R ( ( x  e.  A  |->  C ) `  x ) )
32 fveq2 5757 . . . . 5  |-  ( y  =  x  ->  (
( x  e.  A  |->  B ) `  y
)  =  ( ( x  e.  A  |->  B ) `  x ) )
33 fveq2 5757 . . . . 5  |-  ( y  =  x  ->  (
( x  e.  A  |->  C ) `  y
)  =  ( ( x  e.  A  |->  C ) `  x ) )
3432, 33oveq12d 6128 . . . 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
) ) )
3526, 5, 30, 31, 34cbvmptf 24099 . . 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
) ) )
36 simpr 449 . . . . . 6  |-  ( (
ph  /\  x  e.  A )  ->  x  e.  A )
375fvmpt2f 24103 . . . . . 6  |-  ( ( x  e.  A  /\  B  e.  W )  ->  ( ( x  e.  A  |->  B ) `  x )  =  B )
3836, 2, 37syl2anc 644 . . . . 5  |-  ( (
ph  /\  x  e.  A )  ->  (
( x  e.  A  |->  B ) `  x
)  =  B )
395fvmpt2f 24103 . . . . . 6  |-  ( ( x  e.  A  /\  C  e.  X )  ->  ( ( x  e.  A  |->  C ) `  x )  =  C )
4036, 11, 39syl2anc 644 . . . . 5  |-  ( (
ph  /\  x  e.  A )  ->  (
( x  e.  A  |->  C ) `  x
)  =  C )
4138, 40oveq12d 6128 . . . 4  |-  ( (
ph  /\  x  e.  A )  ->  (
( ( x  e.  A  |->  B ) `  x ) R ( ( x  e.  A  |->  C ) `  x
) )  =  ( B R C ) )
421, 41mpteq2da 4319 . . 3  |-  ( ph  ->  ( x  e.  A  |->  ( ( ( x  e.  A  |->  B ) `
 x ) R ( ( x  e.  A  |->  C ) `  x ) ) )  =  ( x  e.  A  |->  ( B R C ) ) )
4335, 42syl5eq 2486 . 2  |-  ( ph  ->  ( y  e.  A  |->  ( ( ( x  e.  A  |->  B ) `
 y ) R ( ( x  e.  A  |->  C ) `  y ) ) )  =  ( x  e.  A  |->  ( B R C ) ) )
4425, 43eqtrd 2474 1  |-  ( ph  ->  ( F  o F R G )  =  ( x  e.  A  |->  ( B R C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360   F/wnf 1554    = wceq 1653    e. wcel 1727   F/_wnfc 2565   A.wral 2711    e. cmpt 4291    Fn wfn 5478   ` cfv 5483  (class class class)co 6110    o Fcof 6332
This theorem is referenced by:  esumaddf  24484
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-rep 4345  ax-sep 4355  ax-nul 4363  ax-pr 4432
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-reu 2718  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-sn 3844  df-pr 3845  df-op 3847  df-uni 4040  df-iun 4119  df-br 4238  df-opab 4292  df-mpt 4293  df-id 4527  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490  df-fv 5491  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-of 6334
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