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Theorem ofpreima 24034
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  o F 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 4258 . . . . . . 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 2405 . . . . . . 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 6137 . . . . . . . 8  |-  ( R  Fn  ( B  X.  C )  <->  R  =  ( x  e.  B ,  y  e.  C  |->  ( x R y ) ) )
86, 7sylib 189 . . . . . . 7  |-  ( ph  ->  R  =  ( x  e.  B ,  y  e.  C  |->  ( x R y ) ) )
91, 2, 3, 4, 5, 8ofoprabco 24032 . . . . . 6  |-  ( ph  ->  ( F  o F R G )  =  ( R  o.  (
s  e.  A  |->  <.
( F `  s
) ,  ( G `
 s ) >.
) ) )
109cnveqd 5007 . . . . 5  |-  ( ph  ->  `' ( F  o F R G )  =  `' ( R  o.  ( s  e.  A  |-> 
<. ( F `  s
) ,  ( G `
 s ) >.
) ) )
11 cnvco 5015 . . . . 5  |-  `' ( R  o.  ( s  e.  A  |->  <. ( F `  s ) ,  ( G `  s ) >. )
)  =  ( `' ( s  e.  A  |-> 
<. ( F `  s
) ,  ( G `
 s ) >.
)  o.  `' R
)
1210, 11syl6eq 2452 . . . 4  |-  ( ph  ->  `' ( F  o F R G )  =  ( `' ( s  e.  A  |->  <. ( F `  s ) ,  ( G `  s ) >. )  o.  `' R ) )
1312imaeq1d 5161 . . 3  |-  ( ph  ->  ( `' ( F  o F R G ) " D )  =  ( ( `' ( s  e.  A  |-> 
<. ( F `  s
) ,  ( G `
 s ) >.
)  o.  `' R
) " D ) )
14 imaco 5334 . . 3  |-  ( ( `' ( s  e.  A  |->  <. ( F `  s ) ,  ( G `  s )
>. )  o.  `' R ) " D
)  =  ( `' ( s  e.  A  |-> 
<. ( F `  s
) ,  ( G `
 s ) >.
) " ( `' R " D ) )
1513, 14syl6eq 2452 . 2  |-  ( ph  ->  ( `' ( F  o F R G ) " D )  =  ( `' ( s  e.  A  |->  <.
( F `  s
) ,  ( G `
 s ) >.
) " ( `' R " D ) ) )
16 dfima2 5164 . . 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 2919 . . . . . . . 8  |-  p  e. 
_V
18 vex 2919 . . . . . . . 8  |-  q  e. 
_V
1917, 18brcnv 5014 . . . . . . 7  |-  ( p `' ( s  e.  A  |->  <. ( F `  s ) ,  ( G `  s )
>. ) q  <->  q (
s  e.  A  |->  <.
( F `  s
) ,  ( G `
 s ) >.
) p )
20 funmpt 5448 . . . . . . . . 9  |-  Fun  (
s  e.  A  |->  <.
( F `  s
) ,  ( G `
 s ) >.
)
21 funbrfv2b 5730 . . . . . . . . 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 8 . . . . . . . 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 4387 . . . . . . . . . . 11  |-  <. ( F `  s ) ,  ( G `  s ) >.  e.  _V
24 eqid 2404 . . . . . . . . . . 11  |-  ( s  e.  A  |->  <. ( F `  s ) ,  ( G `  s ) >. )  =  ( s  e.  A  |->  <. ( F `  s ) ,  ( G `  s )
>. )
2523, 24dmmpti 5533 . . . . . . . . . 10  |-  dom  (
s  e.  A  |->  <.
( F `  s
) ,  ( G `
 s ) >.
)  =  A
2625eleq2i 2468 . . . . . . . . 9  |-  ( q  e.  dom  ( s  e.  A  |->  <. ( F `  s ) ,  ( G `  s ) >. )  <->  q  e.  A )
2726anbi1i 677 . . . . . . . 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 241 . . . . . . 7  |-  ( q ( s  e.  A  |-> 
<. ( F `  s
) ,  ( G `
 s ) >.
) p  <->  ( q  e.  A  /\  (
( s  e.  A  |-> 
<. ( F `  s
) ,  ( G `
 s ) >.
) `  q )  =  p ) )
29 fveq2 5687 . . . . . . . . . . 11  |-  ( s  =  q  ->  ( F `  s )  =  ( F `  q ) )
30 fveq2 5687 . . . . . . . . . . 11  |-  ( s  =  q  ->  ( G `  s )  =  ( G `  q ) )
3129, 30opeq12d 3952 . . . . . . . . . 10  |-  ( s  =  q  ->  <. ( F `  s ) ,  ( G `  s ) >.  =  <. ( F `  q ) ,  ( G `  q ) >. )
32 opex 4387 . . . . . . . . . 10  |-  <. ( F `  q ) ,  ( G `  q ) >.  e.  _V
3331, 24, 32fvmpt 5765 . . . . . . . . 9  |-  ( q  e.  A  ->  (
( s  e.  A  |-> 
<. ( F `  s
) ,  ( G `
 s ) >.
) `  q )  =  <. ( F `  q ) ,  ( G `  q )
>. )
3433eqeq1d 2412 . . . . . . . 8  |-  ( q  e.  A  ->  (
( ( s  e.  A  |->  <. ( F `  s ) ,  ( G `  s )
>. ) `  q )  =  p  <->  <. ( F `
 q ) ,  ( G `  q
) >.  =  p ) )
3534pm5.32i 619 . . . . . . 7  |-  ( ( q  e.  A  /\  ( ( s  e.  A  |->  <. ( F `  s ) ,  ( G `  s )
>. ) `  q )  =  p )  <->  ( q  e.  A  /\  <. ( F `  q ) ,  ( G `  q ) >.  =  p ) )
3619, 28, 353bitri 263 . . . . . 6  |-  ( p `' ( s  e.  A  |->  <. ( F `  s ) ,  ( G `  s )
>. ) q  <->  ( q  e.  A  /\  <. ( F `  q ) ,  ( G `  q ) >.  =  p ) )
3736rexbii 2691 . . . . 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 2516 . . . 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 1626 . . . . 5  |-  F/ q
ph
40 nfab1 2542 . . . . 5  |-  F/_ q { q  |  E. p  e.  ( `' R " D ) ( q  e.  A  /\  <.
( F `  q
) ,  ( G `
 q ) >.  =  p ) }
41 nfcv 2540 . . . . 5  |-  F/_ q U_ p  e.  ( `' R " D ) ( ( `' F " { ( 1st `  p
) } )  i^i  ( `' G " { ( 2nd `  p
) } ) )
42 eliun 4057 . . . . . 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 3490 . . . . . . . . . . 11  |-  ( q  e.  ( ( `' F " { ( 1st `  p ) } )  i^i  ( `' G " { ( 2nd `  p ) } ) )  <->  ( q  e.  ( `' F " { ( 1st `  p
) } )  /\  q  e.  ( `' G " { ( 2nd `  p ) } ) ) )
44 ffn 5550 . . . . . . . . . . . . . 14  |-  ( F : A --> B  ->  F  Fn  A )
45 fniniseg 5810 . . . . . . . . . . . . . 14  |-  ( F  Fn  A  ->  (
q  e.  ( `' F " { ( 1st `  p ) } )  <->  ( q  e.  A  /\  ( F `  q )  =  ( 1st `  p
) ) ) )
462, 44, 453syl 19 . . . . . . . . . . . . 13  |-  ( ph  ->  ( q  e.  ( `' F " { ( 1st `  p ) } )  <->  ( q  e.  A  /\  ( F `  q )  =  ( 1st `  p
) ) ) )
47 ffn 5550 . . . . . . . . . . . . . 14  |-  ( G : A --> C  ->  G  Fn  A )
48 fniniseg 5810 . . . . . . . . . . . . . 14  |-  ( G  Fn  A  ->  (
q  e.  ( `' G " { ( 2nd `  p ) } )  <->  ( q  e.  A  /\  ( G `  q )  =  ( 2nd `  p
) ) ) )
493, 47, 483syl 19 . . . . . . . . . . . . 13  |-  ( ph  ->  ( q  e.  ( `' G " { ( 2nd `  p ) } )  <->  ( q  e.  A  /\  ( G `  q )  =  ( 2nd `  p
) ) ) )
5046, 49anbi12d 692 . . . . . . . . . . . 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 802 . . . . . . . . . . . 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 255 . . . . . . . . . . 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 249 . . . . . . . . . 10  |-  ( ph  ->  ( q  e.  ( ( `' F " { ( 1st `  p
) } )  i^i  ( `' G " { ( 2nd `  p
) } ) )  <-> 
( q  e.  A  /\  ( ( F `  q )  =  ( 1st `  p )  /\  ( G `  q )  =  ( 2nd `  p ) ) ) ) )
5453adantr 452 . . . . . . . . 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 5183 . . . . . . . . . . . . . 14  |-  ( `' R " D ) 
C_  dom  R
56 fndm 5503 . . . . . . . . . . . . . . 15  |-  ( R  Fn  ( B  X.  C )  ->  dom  R  =  ( B  X.  C ) )
576, 56syl 16 . . . . . . . . . . . . . 14  |-  ( ph  ->  dom  R  =  ( B  X.  C ) )
5855, 57syl5sseq 3356 . . . . . . . . . . . . 13  |-  ( ph  ->  ( `' R " D )  C_  ( B  X.  C ) )
5958sselda 3308 . . . . . . . . . . . 12  |-  ( (
ph  /\  p  e.  ( `' R " D ) )  ->  p  e.  ( B  X.  C
) )
60 1st2nd2 6345 . . . . . . . . . . . 12  |-  ( p  e.  ( B  X.  C )  ->  p  =  <. ( 1st `  p
) ,  ( 2nd `  p ) >. )
61 eqeq2 2413 . . . . . . . . . . . 12  |-  ( p  =  <. ( 1st `  p
) ,  ( 2nd `  p ) >.  ->  ( <. ( F `  q
) ,  ( G `
 q ) >.  =  p  <->  <. ( F `  q ) ,  ( G `  q )
>.  =  <. ( 1st `  p ) ,  ( 2nd `  p )
>. ) )
6259, 60, 613syl 19 . . . . . . . . . . 11  |-  ( (
ph  /\  p  e.  ( `' R " D ) )  ->  ( <. ( F `  q ) ,  ( G `  q ) >.  =  p  <->  <. ( F `  q
) ,  ( G `
 q ) >.  =  <. ( 1st `  p
) ,  ( 2nd `  p ) >. )
)
63 fvex 5701 . . . . . . . . . . . 12  |-  ( F `
 q )  e. 
_V
64 fvex 5701 . . . . . . . . . . . 12  |-  ( G `
 q )  e. 
_V
6563, 64opth 4395 . . . . . . . . . . 11  |-  ( <.
( F `  q
) ,  ( G `
 q ) >.  =  <. ( 1st `  p
) ,  ( 2nd `  p ) >.  <->  ( ( F `  q )  =  ( 1st `  p
)  /\  ( G `  q )  =  ( 2nd `  p ) ) )
6662, 65syl6bb 253 . . . . . . . . . 10  |-  ( (
ph  /\  p  e.  ( `' R " D ) )  ->  ( <. ( F `  q ) ,  ( G `  q ) >.  =  p  <-> 
( ( F `  q )  =  ( 1st `  p )  /\  ( G `  q )  =  ( 2nd `  p ) ) ) )
6766anbi2d 685 . . . . . . . . 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 248 . . . . . . . 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 2683 . . . . . . 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 2392 . . . . . . 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 255 . . . . . 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 250 . . . . 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 3326 . . . 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 2448 . . 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 2448 . 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 2436 1  |-  ( ph  ->  ( `' ( F  o F R G ) " D )  =  U_ p  e.  ( `' R " D ) ( ( `' F " { ( 1st `  p ) } )  i^i  ( `' G " { ( 2nd `  p ) } ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721   {cab 2390   E.wrex 2667    i^i cin 3279   {csn 3774   <.cop 3777   U_ciun 4053   class class class wbr 4172    e. cmpt 4226    X. cxp 4835   `'ccnv 4836   dom cdm 4837   "cima 4840    o. ccom 4841   Fun wfun 5407    Fn wfn 5408   -->wf 5409   ` cfv 5413  (class class class)co 6040    e. cmpt2 6042    o Fcof 6262   1stc1st 6306   2ndc2nd 6307
This theorem is referenced by:  sibfof  24607
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-13 1723  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-pow 4337  ax-pr 4363  ax-un 4660
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  df-1st 6308  df-2nd 6309
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