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Theorem offval2f 24068
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 424 . . . . . 6  |-  ( ph  ->  ( x  e.  A  ->  B  e.  W ) )
41, 3ralrimi 2779 . . . . 5  |-  ( ph  ->  A. x  e.  A  B  e.  W )
5 offval2f.a . . . . . 6  |-  F/_ x A
65fnmptf 24062 . . . . 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 5527 . . . 4  |-  ( ph  ->  ( F  Fn  A  <->  ( x  e.  A  |->  B )  Fn  A ) )
107, 9mpbird 224 . . 3  |-  ( ph  ->  F  Fn  A )
11 offval2f.3 . . . . . . 7  |-  ( (
ph  /\  x  e.  A )  ->  C  e.  X )
1211ex 424 . . . . . 6  |-  ( ph  ->  ( x  e.  A  ->  C  e.  X ) )
131, 12ralrimi 2779 . . . . 5  |-  ( ph  ->  A. x  e.  A  C  e.  X )
145fnmptf 24062 . . . . 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 5527 . . . 4  |-  ( ph  ->  ( G  Fn  A  <->  ( x  e.  A  |->  C )  Fn  A ) )
1815, 17mpbird 224 . . 3  |-  ( ph  ->  G  Fn  A )
19 offval2f.1 . . 3  |-  ( ph  ->  A  e.  V )
20 inidm 3542 . . 3  |-  ( A  i^i  A )  =  A
218adantr 452 . . . 4  |-  ( (
ph  /\  y  e.  A )  ->  F  =  ( x  e.  A  |->  B ) )
2221fveq1d 5721 . . 3  |-  ( (
ph  /\  y  e.  A )  ->  ( F `  y )  =  ( ( x  e.  A  |->  B ) `
 y ) )
2316adantr 452 . . . 4  |-  ( (
ph  /\  y  e.  A )  ->  G  =  ( x  e.  A  |->  C ) )
2423fveq1d 5721 . . 3  |-  ( (
ph  /\  y  e.  A )  ->  ( G `  y )  =  ( ( x  e.  A  |->  C ) `
 y ) )
2510, 18, 19, 19, 20, 22, 24offval 6303 . 2  |-  ( ph  ->  ( F  o F R G )  =  ( y  e.  A  |->  ( ( ( x  e.  A  |->  B ) `
 y ) R ( ( x  e.  A  |->  C ) `  y ) ) ) )
26 nfcv 2571 . . . 4  |-  F/_ y A
27 nffvmpt1 5727 . . . . 5  |-  F/_ x
( ( x  e.  A  |->  B ) `  y )
28 nfcv 2571 . . . . 5  |-  F/_ x R
29 nffvmpt1 5727 . . . . 5  |-  F/_ x
( ( x  e.  A  |->  C ) `  y )
3027, 28, 29nfov 6095 . . . 4  |-  F/_ x
( ( ( x  e.  A  |->  B ) `
 y ) R ( ( x  e.  A  |->  C ) `  y ) )
31 nfcv 2571 . . . 4  |-  F/_ y
( ( ( x  e.  A  |->  B ) `
 x ) R ( ( x  e.  A  |->  C ) `  x ) )
32 fveq2 5719 . . . . 5  |-  ( y  =  x  ->  (
( x  e.  A  |->  B ) `  y
)  =  ( ( x  e.  A  |->  B ) `  x ) )
33 fveq2 5719 . . . . 5  |-  ( y  =  x  ->  (
( x  e.  A  |->  C ) `  y
)  =  ( ( x  e.  A  |->  C ) `  x ) )
3432, 33oveq12d 6090 . . . 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 24056 . . 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 448 . . . . . 6  |-  ( (
ph  /\  x  e.  A )  ->  x  e.  A )
375fvmpt2f 24060 . . . . . 6  |-  ( ( x  e.  A  /\  B  e.  W )  ->  ( ( x  e.  A  |->  B ) `  x )  =  B )
3836, 2, 37syl2anc 643 . . . . 5  |-  ( (
ph  /\  x  e.  A )  ->  (
( x  e.  A  |->  B ) `  x
)  =  B )
395fvmpt2f 24060 . . . . . 6  |-  ( ( x  e.  A  /\  C  e.  X )  ->  ( ( x  e.  A  |->  C ) `  x )  =  C )
4036, 11, 39syl2anc 643 . . . . 5  |-  ( (
ph  /\  x  e.  A )  ->  (
( x  e.  A  |->  C ) `  x
)  =  C )
4138, 40oveq12d 6090 . . . 4  |-  ( (
ph  /\  x  e.  A )  ->  (
( ( x  e.  A  |->  B ) `  x ) R ( ( x  e.  A  |->  C ) `  x
) )  =  ( B R C ) )
421, 41mpteq2da 4286 . . 3  |-  ( ph  ->  ( x  e.  A  |->  ( ( ( x  e.  A  |->  B ) `
 x ) R ( ( x  e.  A  |->  C ) `  x ) ) )  =  ( x  e.  A  |->  ( B R C ) ) )
4335, 42syl5eq 2479 . 2  |-  ( ph  ->  ( y  e.  A  |->  ( ( ( x  e.  A  |->  B ) `
 y ) R ( ( x  e.  A  |->  C ) `  y ) ) )  =  ( x  e.  A  |->  ( B R C ) ) )
4425, 43eqtrd 2467 1  |-  ( ph  ->  ( F  o F R G )  =  ( x  e.  A  |->  ( B R C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359   F/wnf 1553    = wceq 1652    e. wcel 1725   F/_wnfc 2558   A.wral 2697    e. cmpt 4258    Fn wfn 5440   ` cfv 5445  (class class class)co 6072    o Fcof 6294
This theorem is referenced by:  esumaddf  24441
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-ov 6075  df-oprab 6076  df-mpt2 6077  df-of 6296
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