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Theorem ofpreima 27734
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 4528 . . . . . . 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 2455 . . . . . . 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 6383 . . . . . . . 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 27732 . . . . . 6  |-  ( ph  ->  ( F  oF R G )  =  ( R  o.  (
s  e.  A  |->  <.
( F `  s
) ,  ( G `
 s ) >.
) ) )
109cnveqd 5167 . . . . 5  |-  ( ph  ->  `' ( F  oF R G )  =  `' ( R  o.  ( s  e.  A  |->  <. ( F `  s ) ,  ( G `  s )
>. ) ) )
11 cnvco 5177 . . . . 5  |-  `' ( R  o.  ( s  e.  A  |->  <. ( F `  s ) ,  ( G `  s ) >. )
)  =  ( `' ( s  e.  A  |-> 
<. ( F `  s
) ,  ( G `
 s ) >.
)  o.  `' R
)
1210, 11syl6eq 2511 . . . 4  |-  ( ph  ->  `' ( F  oF R G )  =  ( `' ( s  e.  A  |->  <.
( F `  s
) ,  ( G `
 s ) >.
)  o.  `' R
) )
1312imaeq1d 5324 . . 3  |-  ( ph  ->  ( `' ( F  oF R G ) " D )  =  ( ( `' ( s  e.  A  |-> 
<. ( F `  s
) ,  ( G `
 s ) >.
)  o.  `' R
) " D ) )
14 imaco 5495 . . 3  |-  ( ( `' ( s  e.  A  |->  <. ( F `  s ) ,  ( G `  s )
>. )  o.  `' R ) " D
)  =  ( `' ( s  e.  A  |-> 
<. ( F `  s
) ,  ( G `
 s ) >.
) " ( `' R " D ) )
1513, 14syl6eq 2511 . 2  |-  ( ph  ->  ( `' ( F  oF R G ) " D )  =  ( `' ( s  e.  A  |->  <.
( F `  s
) ,  ( G `
 s ) >.
) " ( `' R " D ) ) )
16 dfima2 5327 . . 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 3109 . . . . . . . 8  |-  p  e. 
_V
18 vex 3109 . . . . . . . 8  |-  q  e. 
_V
1917, 18brcnv 5174 . . . . . . 7  |-  ( p `' ( s  e.  A  |->  <. ( F `  s ) ,  ( G `  s )
>. ) q  <->  q (
s  e.  A  |->  <.
( F `  s
) ,  ( G `
 s ) >.
) p )
20 funmpt 5606 . . . . . . . . 9  |-  Fun  (
s  e.  A  |->  <.
( F `  s
) ,  ( G `
 s ) >.
)
21 funbrfv2b 5892 . . . . . . . . 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 4701 . . . . . . . . . . 11  |-  <. ( F `  s ) ,  ( G `  s ) >.  e.  _V
24 eqid 2454 . . . . . . . . . . 11  |-  ( s  e.  A  |->  <. ( F `  s ) ,  ( G `  s ) >. )  =  ( s  e.  A  |->  <. ( F `  s ) ,  ( G `  s )
>. )
2523, 24dmmpti 5692 . . . . . . . . . 10  |-  dom  (
s  e.  A  |->  <.
( F `  s
) ,  ( G `
 s ) >.
)  =  A
2625eleq2i 2532 . . . . . . . . 9  |-  ( q  e.  dom  ( s  e.  A  |->  <. ( F `  s ) ,  ( G `  s ) >. )  <->  q  e.  A )
2726anbi1i 693 . . . . . . . 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 5848 . . . . . . . . . . 11  |-  ( s  =  q  ->  ( F `  s )  =  ( F `  q ) )
30 fveq2 5848 . . . . . . . . . . 11  |-  ( s  =  q  ->  ( G `  s )  =  ( G `  q ) )
3129, 30opeq12d 4211 . . . . . . . . . 10  |-  ( s  =  q  ->  <. ( F `  s ) ,  ( G `  s ) >.  =  <. ( F `  q ) ,  ( G `  q ) >. )
32 opex 4701 . . . . . . . . . 10  |-  <. ( F `  q ) ,  ( G `  q ) >.  e.  _V
3331, 24, 32fvmpt 5931 . . . . . . . . 9  |-  ( q  e.  A  ->  (
( s  e.  A  |-> 
<. ( F `  s
) ,  ( G `
 s ) >.
) `  q )  =  <. ( F `  q ) ,  ( G `  q )
>. )
3433eqeq1d 2456 . . . . . . . 8  |-  ( q  e.  A  ->  (
( ( s  e.  A  |->  <. ( F `  s ) ,  ( G `  s )
>. ) `  q )  =  p  <->  <. ( F `
 q ) ,  ( G `  q
) >.  =  p ) )
3534pm5.32i 635 . . . . . . 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 2956 . . . . 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 2588 . . . 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 1712 . . . . 5  |-  F/ q
ph
40 nfab1 2618 . . . . 5  |-  F/_ q { q  |  E. p  e.  ( `' R " D ) ( q  e.  A  /\  <.
( F `  q
) ,  ( G `
 q ) >.  =  p ) }
41 nfcv 2616 . . . . 5  |-  F/_ q U_ p  e.  ( `' R " D ) ( ( `' F " { ( 1st `  p
) } )  i^i  ( `' G " { ( 2nd `  p
) } ) )
42 eliun 4320 . . . . . 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 ffn 5713 . . . . . . . . . . . . 13  |-  ( F : A --> B  ->  F  Fn  A )
44 fniniseg 5984 . . . . . . . . . . . . 13  |-  ( F  Fn  A  ->  (
q  e.  ( `' F " { ( 1st `  p ) } )  <->  ( q  e.  A  /\  ( F `  q )  =  ( 1st `  p
) ) ) )
452, 43, 443syl 20 . . . . . . . . . . . 12  |-  ( ph  ->  ( q  e.  ( `' F " { ( 1st `  p ) } )  <->  ( q  e.  A  /\  ( F `  q )  =  ( 1st `  p
) ) ) )
46 ffn 5713 . . . . . . . . . . . . 13  |-  ( G : A --> C  ->  G  Fn  A )
47 fniniseg 5984 . . . . . . . . . . . . 13  |-  ( G  Fn  A  ->  (
q  e.  ( `' G " { ( 2nd `  p ) } )  <->  ( q  e.  A  /\  ( G `  q )  =  ( 2nd `  p
) ) ) )
483, 46, 473syl 20 . . . . . . . . . . . 12  |-  ( ph  ->  ( q  e.  ( `' G " { ( 2nd `  p ) } )  <->  ( q  e.  A  /\  ( G `  q )  =  ( 2nd `  p
) ) ) )
4945, 48anbi12d 708 . . . . . . . . . . 11  |-  ( 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
) ) ) ) )
50 elin 3673 . . . . . . . . . . 11  |-  ( q  e.  ( ( `' F " { ( 1st `  p ) } )  i^i  ( `' G " { ( 2nd `  p ) } ) )  <->  ( q  e.  ( `' F " { ( 1st `  p
) } )  /\  q  e.  ( `' G " { ( 2nd `  p ) } ) ) )
51 anandi 826 . . . . . . . . . . 11  |-  ( ( 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
) ) ) )
5249, 50, 513bitr4g 288 . . . . . . . . . 10  |-  ( ph  ->  ( q  e.  ( ( `' F " { ( 1st `  p
) } )  i^i  ( `' G " { ( 2nd `  p
) } ) )  <-> 
( q  e.  A  /\  ( ( F `  q )  =  ( 1st `  p )  /\  ( G `  q )  =  ( 2nd `  p ) ) ) ) )
5352adantr 463 . . . . . . . . 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 ) ) ) ) )
54 cnvimass 5345 . . . . . . . . . . . . . 14  |-  ( `' R " D ) 
C_  dom  R
55 fndm 5662 . . . . . . . . . . . . . . 15  |-  ( R  Fn  ( B  X.  C )  ->  dom  R  =  ( B  X.  C ) )
566, 55syl 16 . . . . . . . . . . . . . 14  |-  ( ph  ->  dom  R  =  ( B  X.  C ) )
5754, 56syl5sseq 3537 . . . . . . . . . . . . 13  |-  ( ph  ->  ( `' R " D )  C_  ( B  X.  C ) )
5857sselda 3489 . . . . . . . . . . . 12  |-  ( (
ph  /\  p  e.  ( `' R " D ) )  ->  p  e.  ( B  X.  C
) )
59 1st2nd2 6810 . . . . . . . . . . . 12  |-  ( p  e.  ( B  X.  C )  ->  p  =  <. ( 1st `  p
) ,  ( 2nd `  p ) >. )
60 eqeq2 2469 . . . . . . . . . . . 12  |-  ( p  =  <. ( 1st `  p
) ,  ( 2nd `  p ) >.  ->  ( <. ( F `  q
) ,  ( G `
 q ) >.  =  p  <->  <. ( F `  q ) ,  ( G `  q )
>.  =  <. ( 1st `  p ) ,  ( 2nd `  p )
>. ) )
6158, 59, 603syl 20 . . . . . . . . . . 11  |-  ( (
ph  /\  p  e.  ( `' R " D ) )  ->  ( <. ( F `  q ) ,  ( G `  q ) >.  =  p  <->  <. ( F `  q
) ,  ( G `
 q ) >.  =  <. ( 1st `  p
) ,  ( 2nd `  p ) >. )
)
62 fvex 5858 . . . . . . . . . . . 12  |-  ( F `
 q )  e. 
_V
63 fvex 5858 . . . . . . . . . . . 12  |-  ( G `
 q )  e. 
_V
6462, 63opth 4711 . . . . . . . . . . 11  |-  ( <.
( F `  q
) ,  ( G `
 q ) >.  =  <. ( 1st `  p
) ,  ( 2nd `  p ) >.  <->  ( ( F `  q )  =  ( 1st `  p
)  /\  ( G `  q )  =  ( 2nd `  p ) ) )
6561, 64syl6bb 261 . . . . . . . . . 10  |-  ( (
ph  /\  p  e.  ( `' R " D ) )  ->  ( <. ( F `  q ) ,  ( G `  q ) >.  =  p  <-> 
( ( F `  q )  =  ( 1st `  p )  /\  ( G `  q )  =  ( 2nd `  p ) ) ) )
6665anbi2d 701 . . . . . . . . 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
) ) ) ) )
6753, 66bitr4d 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 )
) )
6867rexbidva 2962 . . . . . . 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 )
) )
69 abid 2441 . . . . . . 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 )
)
7068, 69syl6bbr 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 ) } ) )
7142, 70syl5rbb 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
) } ) ) ) )
7239, 40, 41, 71eqrd 3507 . . . 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
) } ) ) )
7338, 72syl5eq 2507 . . 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
) } ) ) )
7416, 73syl5eq 2507 . 2  |-  ( ph  ->  ( `' ( s  e.  A  |->  <. ( F `  s ) ,  ( G `  s ) >. ) " ( `' R " D ) )  = 
U_ p  e.  ( `' R " D ) ( ( `' F " { ( 1st `  p
) } )  i^i  ( `' G " { ( 2nd `  p
) } ) ) )
7515, 74eqtrd 2495 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 367    = wceq 1398    e. wcel 1823   {cab 2439   E.wrex 2805    i^i cin 3460   {csn 4016   <.cop 4022   U_ciun 4315   class class class wbr 4439    |-> cmpt 4497    X. cxp 4986   `'ccnv 4987   dom cdm 4988   "cima 4991    o. ccom 4992   Fun wfun 5564    Fn wfn 5565   -->wf 5566   ` cfv 5570  (class class class)co 6270    |-> cmpt2 6272    oFcof 6511   1stc1st 6771   2ndc2nd 6772
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-of 6513  df-1st 6773  df-2nd 6774
This theorem is referenced by:  ofpreima2  27735
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