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Theorem offval2f 24033
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 2747 . . . . 5  |-  ( ph  ->  A. x  e.  A  B  e.  W )
5 offval2f.a . . . . . 6  |-  F/_ x A
65fnmptf 24027 . . . . 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 5495 . . . 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 2747 . . . . 5  |-  ( ph  ->  A. x  e.  A  C  e.  X )
145fnmptf 24027 . . . . 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 5495 . . . 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 3510 . . 3  |-  ( A  i^i  A )  =  A
218adantr 452 . . . 4  |-  ( (
ph  /\  y  e.  A )  ->  F  =  ( x  e.  A  |->  B ) )
2221fveq1d 5689 . . 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 5689 . . 3  |-  ( (
ph  /\  y  e.  A )  ->  ( G `  y )  =  ( ( x  e.  A  |->  C ) `
 y ) )
2510, 18, 19, 19, 20, 22, 24offval 6271 . 2  |-  ( ph  ->  ( F  o F R G )  =  ( y  e.  A  |->  ( ( ( x  e.  A  |->  B ) `
 y ) R ( ( x  e.  A  |->  C ) `  y ) ) ) )
26 nfcv 2540 . . . 4  |-  F/_ y A
27 nffvmpt1 5695 . . . . 5  |-  F/_ x
( ( x  e.  A  |->  B ) `  y )
28 nfcv 2540 . . . . 5  |-  F/_ x R
29 nffvmpt1 5695 . . . . 5  |-  F/_ x
( ( x  e.  A  |->  C ) `  y )
3027, 28, 29nfov 6063 . . . 4  |-  F/_ x
( ( ( x  e.  A  |->  B ) `
 y ) R ( ( x  e.  A  |->  C ) `  y ) )
31 nfcv 2540 . . . 4  |-  F/_ y
( ( ( x  e.  A  |->  B ) `
 x ) R ( ( x  e.  A  |->  C ) `  x ) )
32 fveq2 5687 . . . . 5  |-  ( y  =  x  ->  (
( x  e.  A  |->  B ) `  y
)  =  ( ( x  e.  A  |->  B ) `  x ) )
33 fveq2 5687 . . . . 5  |-  ( y  =  x  ->  (
( x  e.  A  |->  C ) `  y
)  =  ( ( x  e.  A  |->  C ) `  x ) )
3432, 33oveq12d 6058 . . . 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 24021 . . 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 24025 . . . . . 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 24025 . . . . . 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 6058 . . . 4  |-  ( (
ph  /\  x  e.  A )  ->  (
( ( x  e.  A  |->  B ) `  x ) R ( ( x  e.  A  |->  C ) `  x
) )  =  ( B R C ) )
421, 41mpteq2da 4254 . . 3  |-  ( ph  ->  ( x  e.  A  |->  ( ( ( x  e.  A  |->  B ) `
 x ) R ( ( x  e.  A  |->  C ) `  x ) ) )  =  ( x  e.  A  |->  ( B R C ) ) )
4335, 42syl5eq 2448 . 2  |-  ( ph  ->  ( y  e.  A  |->  ( ( ( x  e.  A  |->  B ) `
 y ) R ( ( x  e.  A  |->  C ) `  y ) ) )  =  ( x  e.  A  |->  ( B R C ) ) )
4425, 43eqtrd 2436 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 1550    = wceq 1649    e. wcel 1721   F/_wnfc 2527   A.wral 2666    e. cmpt 4226    Fn wfn 5408   ` cfv 5413  (class class class)co 6040    o Fcof 6262
This theorem is referenced by:  esumaddf  24406
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264
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