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Theorem ofpreima 25989
Description: Express the preimage of a function operation as a union of preimages. (Contributed by Thierry Arnoux, 8-Mar-2018.)
Hypotheses
Ref Expression
ofpreima.1  |-  ( ph  ->  F : A --> B )
ofpreima.2  |-  ( ph  ->  G : A --> C )
ofpreima.3  |-  ( ph  ->  A  e.  V )
ofpreima.4  |-  ( ph  ->  R  Fn  ( B  X.  C ) )
Assertion
Ref Expression
ofpreima  |-  ( ph  ->  ( `' ( F  oF R G ) " D )  =  U_ p  e.  ( `' R " D ) ( ( `' F " { ( 1st `  p ) } )  i^i  ( `' G " { ( 2nd `  p ) } ) ) )
Distinct variable groups:    A, p    D, p    F, p    G, p    R, p    ph, p
Allowed substitution hints:    B( p)    C( p)    V( p)

Proof of Theorem ofpreima
Dummy variables  q 
s  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nfmpt1 4386 . . . . . . 7  |-  F/_ s
( s  e.  A  |-> 
<. ( F `  s
) ,  ( G `
 s ) >.
)
2 ofpreima.1 . . . . . . 7  |-  ( ph  ->  F : A --> B )
3 ofpreima.2 . . . . . . 7  |-  ( ph  ->  G : A --> C )
4 ofpreima.3 . . . . . . 7  |-  ( ph  ->  A  e.  V )
5 eqidd 2444 . . . . . . 7  |-  ( ph  ->  ( s  e.  A  |-> 
<. ( F `  s
) ,  ( G `
 s ) >.
)  =  ( s  e.  A  |->  <. ( F `  s ) ,  ( G `  s ) >. )
)
6 ofpreima.4 . . . . . . . 8  |-  ( ph  ->  R  Fn  ( B  X.  C ) )
7 fnov 6203 . . . . . . . 8  |-  ( R  Fn  ( B  X.  C )  <->  R  =  ( x  e.  B ,  y  e.  C  |->  ( x R y ) ) )
86, 7sylib 196 . . . . . . 7  |-  ( ph  ->  R  =  ( x  e.  B ,  y  e.  C  |->  ( x R y ) ) )
91, 2, 3, 4, 5, 8ofoprabco 25987 . . . . . 6  |-  ( ph  ->  ( F  oF R G )  =  ( R  o.  (
s  e.  A  |->  <.
( F `  s
) ,  ( G `
 s ) >.
) ) )
109cnveqd 5020 . . . . 5  |-  ( ph  ->  `' ( F  oF R G )  =  `' ( R  o.  ( s  e.  A  |->  <. ( F `  s ) ,  ( G `  s )
>. ) ) )
11 cnvco 5030 . . . . 5  |-  `' ( R  o.  ( s  e.  A  |->  <. ( F `  s ) ,  ( G `  s ) >. )
)  =  ( `' ( s  e.  A  |-> 
<. ( F `  s
) ,  ( G `
 s ) >.
)  o.  `' R
)
1210, 11syl6eq 2491 . . . 4  |-  ( ph  ->  `' ( F  oF R G )  =  ( `' ( s  e.  A  |->  <.
( F `  s
) ,  ( G `
 s ) >.
)  o.  `' R
) )
1312imaeq1d 5173 . . 3  |-  ( ph  ->  ( `' ( F  oF R G ) " D )  =  ( ( `' ( s  e.  A  |-> 
<. ( F `  s
) ,  ( G `
 s ) >.
)  o.  `' R
) " D ) )
14 imaco 5348 . . 3  |-  ( ( `' ( s  e.  A  |->  <. ( F `  s ) ,  ( G `  s )
>. )  o.  `' R ) " D
)  =  ( `' ( s  e.  A  |-> 
<. ( F `  s
) ,  ( G `
 s ) >.
) " ( `' R " D ) )
1513, 14syl6eq 2491 . 2  |-  ( ph  ->  ( `' ( F  oF R G ) " D )  =  ( `' ( s  e.  A  |->  <.
( F `  s
) ,  ( G `
 s ) >.
) " ( `' R " D ) ) )
16 dfima2 5176 . . 3  |-  ( `' ( s  e.  A  |-> 
<. ( F `  s
) ,  ( G `
 s ) >.
) " ( `' R " D ) )  =  { q  |  E. p  e.  ( `' R " D ) p `' ( s  e.  A  |-> 
<. ( F `  s
) ,  ( G `
 s ) >.
) q }
17 vex 2980 . . . . . . . 8  |-  p  e. 
_V
18 vex 2980 . . . . . . . 8  |-  q  e. 
_V
1917, 18brcnv 5027 . . . . . . 7  |-  ( p `' ( s  e.  A  |->  <. ( F `  s ) ,  ( G `  s )
>. ) q  <->  q (
s  e.  A  |->  <.
( F `  s
) ,  ( G `
 s ) >.
) p )
20 funmpt 5459 . . . . . . . . 9  |-  Fun  (
s  e.  A  |->  <.
( F `  s
) ,  ( G `
 s ) >.
)
21 funbrfv2b 5741 . . . . . . . . 9  |-  ( Fun  ( s  e.  A  |-> 
<. ( F `  s
) ,  ( G `
 s ) >.
)  ->  ( q
( s  e.  A  |-> 
<. ( F `  s
) ,  ( G `
 s ) >.
) p  <->  ( q  e.  dom  ( s  e.  A  |->  <. ( F `  s ) ,  ( G `  s )
>. )  /\  (
( s  e.  A  |-> 
<. ( F `  s
) ,  ( G `
 s ) >.
) `  q )  =  p ) ) )
2220, 21ax-mp 5 . . . . . . . 8  |-  ( q ( s  e.  A  |-> 
<. ( F `  s
) ,  ( G `
 s ) >.
) p  <->  ( q  e.  dom  ( s  e.  A  |->  <. ( F `  s ) ,  ( G `  s )
>. )  /\  (
( s  e.  A  |-> 
<. ( F `  s
) ,  ( G `
 s ) >.
) `  q )  =  p ) )
23 opex 4561 . . . . . . . . . . 11  |-  <. ( F `  s ) ,  ( G `  s ) >.  e.  _V
24 eqid 2443 . . . . . . . . . . 11  |-  ( s  e.  A  |->  <. ( F `  s ) ,  ( G `  s ) >. )  =  ( s  e.  A  |->  <. ( F `  s ) ,  ( G `  s )
>. )
2523, 24dmmpti 5545 . . . . . . . . . 10  |-  dom  (
s  e.  A  |->  <.
( F `  s
) ,  ( G `
 s ) >.
)  =  A
2625eleq2i 2507 . . . . . . . . 9  |-  ( q  e.  dom  ( s  e.  A  |->  <. ( F `  s ) ,  ( G `  s ) >. )  <->  q  e.  A )
2726anbi1i 695 . . . . . . . 8  |-  ( ( q  e.  dom  (
s  e.  A  |->  <.
( F `  s
) ,  ( G `
 s ) >.
)  /\  ( (
s  e.  A  |->  <.
( F `  s
) ,  ( G `
 s ) >.
) `  q )  =  p )  <->  ( q  e.  A  /\  (
( s  e.  A  |-> 
<. ( F `  s
) ,  ( G `
 s ) >.
) `  q )  =  p ) )
2822, 27bitri 249 . . . . . . 7  |-  ( q ( s  e.  A  |-> 
<. ( F `  s
) ,  ( G `
 s ) >.
) p  <->  ( q  e.  A  /\  (
( s  e.  A  |-> 
<. ( F `  s
) ,  ( G `
 s ) >.
) `  q )  =  p ) )
29 fveq2 5696 . . . . . . . . . . 11  |-  ( s  =  q  ->  ( F `  s )  =  ( F `  q ) )
30 fveq2 5696 . . . . . . . . . . 11  |-  ( s  =  q  ->  ( G `  s )  =  ( G `  q ) )
3129, 30opeq12d 4072 . . . . . . . . . 10  |-  ( s  =  q  ->  <. ( F `  s ) ,  ( G `  s ) >.  =  <. ( F `  q ) ,  ( G `  q ) >. )
32 opex 4561 . . . . . . . . . 10  |-  <. ( F `  q ) ,  ( G `  q ) >.  e.  _V
3331, 24, 32fvmpt 5779 . . . . . . . . 9  |-  ( q  e.  A  ->  (
( s  e.  A  |-> 
<. ( F `  s
) ,  ( G `
 s ) >.
) `  q )  =  <. ( F `  q ) ,  ( G `  q )
>. )
3433eqeq1d 2451 . . . . . . . 8  |-  ( q  e.  A  ->  (
( ( s  e.  A  |->  <. ( F `  s ) ,  ( G `  s )
>. ) `  q )  =  p  <->  <. ( F `
 q ) ,  ( G `  q
) >.  =  p ) )
3534pm5.32i 637 . . . . . . 7  |-  ( ( q  e.  A  /\  ( ( s  e.  A  |->  <. ( F `  s ) ,  ( G `  s )
>. ) `  q )  =  p )  <->  ( q  e.  A  /\  <. ( F `  q ) ,  ( G `  q ) >.  =  p ) )
3619, 28, 353bitri 271 . . . . . 6  |-  ( p `' ( s  e.  A  |->  <. ( F `  s ) ,  ( G `  s )
>. ) q  <->  ( q  e.  A  /\  <. ( F `  q ) ,  ( G `  q ) >.  =  p ) )
3736rexbii 2745 . . . . 5  |-  ( E. p  e.  ( `' R " D ) p `' ( s  e.  A  |->  <. ( F `  s ) ,  ( G `  s ) >. )
q  <->  E. p  e.  ( `' R " D ) ( q  e.  A  /\  <. ( F `  q ) ,  ( G `  q )
>.  =  p )
)
3837abbii 2560 . . . 4  |-  { q  |  E. p  e.  ( `' R " D ) p `' ( s  e.  A  |-> 
<. ( F `  s
) ,  ( G `
 s ) >.
) q }  =  { q  |  E. p  e.  ( `' R " D ) ( q  e.  A  /\  <.
( F `  q
) ,  ( G `
 q ) >.  =  p ) }
39 nfv 1673 . . . . 5  |-  F/ q
ph
40 nfab1 2586 . . . . 5  |-  F/_ q { q  |  E. p  e.  ( `' R " D ) ( q  e.  A  /\  <.
( F `  q
) ,  ( G `
 q ) >.  =  p ) }
41 nfcv 2584 . . . . 5  |-  F/_ q U_ p  e.  ( `' R " D ) ( ( `' F " { ( 1st `  p
) } )  i^i  ( `' G " { ( 2nd `  p
) } ) )
42 eliun 4180 . . . . . 6  |-  ( q  e.  U_ p  e.  ( `' R " D ) ( ( `' F " { ( 1st `  p ) } )  i^i  ( `' G " { ( 2nd `  p ) } ) )  <->  E. p  e.  ( `' R " D ) q  e.  ( ( `' F " { ( 1st `  p
) } )  i^i  ( `' G " { ( 2nd `  p
) } ) ) )
43 elin 3544 . . . . . . . . . . 11  |-  ( q  e.  ( ( `' F " { ( 1st `  p ) } )  i^i  ( `' G " { ( 2nd `  p ) } ) )  <->  ( q  e.  ( `' F " { ( 1st `  p
) } )  /\  q  e.  ( `' G " { ( 2nd `  p ) } ) ) )
44 ffn 5564 . . . . . . . . . . . . . 14  |-  ( F : A --> B  ->  F  Fn  A )
45 fniniseg 5829 . . . . . . . . . . . . . 14  |-  ( F  Fn  A  ->  (
q  e.  ( `' F " { ( 1st `  p ) } )  <->  ( q  e.  A  /\  ( F `  q )  =  ( 1st `  p
) ) ) )
462, 44, 453syl 20 . . . . . . . . . . . . 13  |-  ( ph  ->  ( q  e.  ( `' F " { ( 1st `  p ) } )  <->  ( q  e.  A  /\  ( F `  q )  =  ( 1st `  p
) ) ) )
47 ffn 5564 . . . . . . . . . . . . . 14  |-  ( G : A --> C  ->  G  Fn  A )
48 fniniseg 5829 . . . . . . . . . . . . . 14  |-  ( G  Fn  A  ->  (
q  e.  ( `' G " { ( 2nd `  p ) } )  <->  ( q  e.  A  /\  ( G `  q )  =  ( 2nd `  p
) ) ) )
493, 47, 483syl 20 . . . . . . . . . . . . 13  |-  ( ph  ->  ( q  e.  ( `' G " { ( 2nd `  p ) } )  <->  ( q  e.  A  /\  ( G `  q )  =  ( 2nd `  p
) ) ) )
5046, 49anbi12d 710 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( q  e.  ( `' F " { ( 1st `  p
) } )  /\  q  e.  ( `' G " { ( 2nd `  p ) } ) )  <->  ( ( q  e.  A  /\  ( F `  q )  =  ( 1st `  p
) )  /\  (
q  e.  A  /\  ( G `  q )  =  ( 2nd `  p
) ) ) ) )
51 anandi 824 . . . . . . . . . . . 12  |-  ( ( q  e.  A  /\  ( ( F `  q )  =  ( 1st `  p )  /\  ( G `  q )  =  ( 2nd `  p ) ) )  <->  ( (
q  e.  A  /\  ( F `  q )  =  ( 1st `  p
) )  /\  (
q  e.  A  /\  ( G `  q )  =  ( 2nd `  p
) ) ) )
5250, 51syl6bbr 263 . . . . . . . . . . 11  |-  ( ph  ->  ( ( q  e.  ( `' F " { ( 1st `  p
) } )  /\  q  e.  ( `' G " { ( 2nd `  p ) } ) )  <->  ( q  e.  A  /\  ( ( F `  q )  =  ( 1st `  p
)  /\  ( G `  q )  =  ( 2nd `  p ) ) ) ) )
5343, 52syl5bb 257 . . . . . . . . . 10  |-  ( ph  ->  ( q  e.  ( ( `' F " { ( 1st `  p
) } )  i^i  ( `' G " { ( 2nd `  p
) } ) )  <-> 
( q  e.  A  /\  ( ( F `  q )  =  ( 1st `  p )  /\  ( G `  q )  =  ( 2nd `  p ) ) ) ) )
5453adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  p  e.  ( `' R " D ) )  ->  ( q  e.  ( ( `' F " { ( 1st `  p
) } )  i^i  ( `' G " { ( 2nd `  p
) } ) )  <-> 
( q  e.  A  /\  ( ( F `  q )  =  ( 1st `  p )  /\  ( G `  q )  =  ( 2nd `  p ) ) ) ) )
55 cnvimass 5194 . . . . . . . . . . . . . 14  |-  ( `' R " D ) 
C_  dom  R
56 fndm 5515 . . . . . . . . . . . . . . 15  |-  ( R  Fn  ( B  X.  C )  ->  dom  R  =  ( B  X.  C ) )
576, 56syl 16 . . . . . . . . . . . . . 14  |-  ( ph  ->  dom  R  =  ( B  X.  C ) )
5855, 57syl5sseq 3409 . . . . . . . . . . . . 13  |-  ( ph  ->  ( `' R " D )  C_  ( B  X.  C ) )
5958sselda 3361 . . . . . . . . . . . 12  |-  ( (
ph  /\  p  e.  ( `' R " D ) )  ->  p  e.  ( B  X.  C
) )
60 1st2nd2 6618 . . . . . . . . . . . 12  |-  ( p  e.  ( B  X.  C )  ->  p  =  <. ( 1st `  p
) ,  ( 2nd `  p ) >. )
61 eqeq2 2452 . . . . . . . . . . . 12  |-  ( p  =  <. ( 1st `  p
) ,  ( 2nd `  p ) >.  ->  ( <. ( F `  q
) ,  ( G `
 q ) >.  =  p  <->  <. ( F `  q ) ,  ( G `  q )
>.  =  <. ( 1st `  p ) ,  ( 2nd `  p )
>. ) )
6259, 60, 613syl 20 . . . . . . . . . . 11  |-  ( (
ph  /\  p  e.  ( `' R " D ) )  ->  ( <. ( F `  q ) ,  ( G `  q ) >.  =  p  <->  <. ( F `  q
) ,  ( G `
 q ) >.  =  <. ( 1st `  p
) ,  ( 2nd `  p ) >. )
)
63 fvex 5706 . . . . . . . . . . . 12  |-  ( F `
 q )  e. 
_V
64 fvex 5706 . . . . . . . . . . . 12  |-  ( G `
 q )  e. 
_V
6563, 64opth 4571 . . . . . . . . . . 11  |-  ( <.
( F `  q
) ,  ( G `
 q ) >.  =  <. ( 1st `  p
) ,  ( 2nd `  p ) >.  <->  ( ( F `  q )  =  ( 1st `  p
)  /\  ( G `  q )  =  ( 2nd `  p ) ) )
6662, 65syl6bb 261 . . . . . . . . . 10  |-  ( (
ph  /\  p  e.  ( `' R " D ) )  ->  ( <. ( F `  q ) ,  ( G `  q ) >.  =  p  <-> 
( ( F `  q )  =  ( 1st `  p )  /\  ( G `  q )  =  ( 2nd `  p ) ) ) )
6766anbi2d 703 . . . . . . . . 9  |-  ( (
ph  /\  p  e.  ( `' R " D ) )  ->  ( (
q  e.  A  /\  <.
( F `  q
) ,  ( G `
 q ) >.  =  p )  <->  ( q  e.  A  /\  (
( F `  q
)  =  ( 1st `  p )  /\  ( G `  q )  =  ( 2nd `  p
) ) ) ) )
6854, 67bitr4d 256 . . . . . . . 8  |-  ( (
ph  /\  p  e.  ( `' R " D ) )  ->  ( q  e.  ( ( `' F " { ( 1st `  p
) } )  i^i  ( `' G " { ( 2nd `  p
) } ) )  <-> 
( q  e.  A  /\  <. ( F `  q ) ,  ( G `  q )
>.  =  p )
) )
6968rexbidva 2737 . . . . . . 7  |-  ( ph  ->  ( E. p  e.  ( `' R " D ) q  e.  ( ( `' F " { ( 1st `  p
) } )  i^i  ( `' G " { ( 2nd `  p
) } ) )  <->  E. p  e.  ( `' R " D ) ( q  e.  A  /\  <. ( F `  q ) ,  ( G `  q )
>.  =  p )
) )
70 abid 2431 . . . . . . 7  |-  ( q  e.  { q  |  E. p  e.  ( `' R " D ) ( q  e.  A  /\  <. ( F `  q ) ,  ( G `  q )
>.  =  p ) } 
<->  E. p  e.  ( `' R " D ) ( q  e.  A  /\  <. ( F `  q ) ,  ( G `  q )
>.  =  p )
)
7169, 70syl6bbr 263 . . . . . 6  |-  ( ph  ->  ( E. p  e.  ( `' R " D ) q  e.  ( ( `' F " { ( 1st `  p
) } )  i^i  ( `' G " { ( 2nd `  p
) } ) )  <-> 
q  e.  { q  |  E. p  e.  ( `' R " D ) ( q  e.  A  /\  <. ( F `  q ) ,  ( G `  q ) >.  =  p ) } ) )
7242, 71syl5rbb 258 . . . . 5  |-  ( ph  ->  ( q  e.  {
q  |  E. p  e.  ( `' R " D ) ( q  e.  A  /\  <. ( F `  q ) ,  ( G `  q ) >.  =  p ) }  <->  q  e.  U_ p  e.  ( `' R " D ) ( ( `' F " { ( 1st `  p
) } )  i^i  ( `' G " { ( 2nd `  p
) } ) ) ) )
7339, 40, 41, 72eqrd 3379 . . . 4  |-  ( ph  ->  { q  |  E. p  e.  ( `' R " D ) ( q  e.  A  /\  <.
( F `  q
) ,  ( G `
 q ) >.  =  p ) }  =  U_ p  e.  ( `' R " D ) ( ( `' F " { ( 1st `  p
) } )  i^i  ( `' G " { ( 2nd `  p
) } ) ) )
7438, 73syl5eq 2487 . . 3  |-  ( ph  ->  { q  |  E. p  e.  ( `' R " D ) p `' ( s  e.  A  |->  <. ( F `  s ) ,  ( G `  s )
>. ) q }  =  U_ p  e.  ( `' R " D ) ( ( `' F " { ( 1st `  p
) } )  i^i  ( `' G " { ( 2nd `  p
) } ) ) )
7516, 74syl5eq 2487 . 2  |-  ( ph  ->  ( `' ( s  e.  A  |->  <. ( F `  s ) ,  ( G `  s ) >. ) " ( `' R " D ) )  = 
U_ p  e.  ( `' R " D ) ( ( `' F " { ( 1st `  p
) } )  i^i  ( `' G " { ( 2nd `  p
) } ) ) )
7615, 75eqtrd 2475 1  |-  ( ph  ->  ( `' ( F  oF R G ) " D )  =  U_ p  e.  ( `' R " D ) ( ( `' F " { ( 1st `  p ) } )  i^i  ( `' G " { ( 2nd `  p ) } ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   {cab 2429   E.wrex 2721    i^i cin 3332   {csn 3882   <.cop 3888   U_ciun 4176   class class class wbr 4297    e. cmpt 4355    X. cxp 4843   `'ccnv 4844   dom cdm 4845   "cima 4848    o. ccom 4849   Fun wfun 5417    Fn wfn 5418   -->wf 5419   ` cfv 5423  (class class class)co 6096    e. cmpt2 6098    oFcof 6323   1stc1st 6580   2ndc2nd 6581
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-reu 2727  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-of 6325  df-1st 6582  df-2nd 6583
This theorem is referenced by:  ofpreima2  25990
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