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Theorem ofpreima2 26004
Description: Express the preimage of a function operation as a union of preimages. This version of ofpreima 26003 iterates the union over a smaller set. (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
ofpreima2  |-  ( ph  ->  ( `' ( F  oF R G ) " D )  =  U_ p  e.  ( ( `' R " D )  i^i  ( ran  F  X.  ran  G
) ) ( ( `' 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 ofpreima2
StepHypRef Expression
1 ofpreima.1 . . . 4  |-  ( ph  ->  F : A --> B )
2 ofpreima.2 . . . 4  |-  ( ph  ->  G : A --> C )
3 ofpreima.3 . . . 4  |-  ( ph  ->  A  e.  V )
4 ofpreima.4 . . . 4  |-  ( ph  ->  R  Fn  ( B  X.  C ) )
51, 2, 3, 4ofpreima 26003 . . 3  |-  ( ph  ->  ( `' ( F  oF R G ) " D )  =  U_ p  e.  ( `' R " D ) ( ( `' F " { ( 1st `  p ) } )  i^i  ( `' G " { ( 2nd `  p ) } ) ) )
6 inundif 3776 . . . . 5  |-  ( ( ( `' R " D )  i^i  ( ran  F  X.  ran  G
) )  u.  (
( `' R " D )  \  ( ran  F  X.  ran  G
) ) )  =  ( `' R " D )
7 iuneq1 4203 . . . . 5  |-  ( ( ( ( `' R " D )  i^i  ( ran  F  X.  ran  G
) )  u.  (
( `' R " D )  \  ( ran  F  X.  ran  G
) ) )  =  ( `' R " D )  ->  U_ p  e.  ( ( ( `' R " D )  i^i  ( ran  F  X.  ran  G ) )  u.  ( ( `' R " D ) 
\  ( ran  F  X.  ran  G ) ) ) ( ( `' F " { ( 1st `  p ) } )  i^i  ( `' G " { ( 2nd `  p ) } ) )  = 
U_ p  e.  ( `' R " D ) ( ( `' F " { ( 1st `  p
) } )  i^i  ( `' G " { ( 2nd `  p
) } ) ) )
86, 7ax-mp 5 . . . 4  |-  U_ p  e.  ( ( ( `' R " D )  i^i  ( ran  F  X.  ran  G ) )  u.  ( ( `' R " D ) 
\  ( ran  F  X.  ran  G ) ) ) ( ( `' F " { ( 1st `  p ) } )  i^i  ( `' G " { ( 2nd `  p ) } ) )  = 
U_ p  e.  ( `' R " D ) ( ( `' F " { ( 1st `  p
) } )  i^i  ( `' G " { ( 2nd `  p
) } ) )
9 iunxun 4271 . . . 4  |-  U_ p  e.  ( ( ( `' R " D )  i^i  ( ran  F  X.  ran  G ) )  u.  ( ( `' R " D ) 
\  ( ran  F  X.  ran  G ) ) ) ( ( `' F " { ( 1st `  p ) } )  i^i  ( `' G " { ( 2nd `  p ) } ) )  =  ( U_ p  e.  ( ( `' R " D )  i^i  ( ran  F  X.  ran  G
) ) ( ( `' F " { ( 1st `  p ) } )  i^i  ( `' G " { ( 2nd `  p ) } ) )  u. 
U_ p  e.  ( ( `' R " D )  \  ( ran  F  X.  ran  G
) ) ( ( `' F " { ( 1st `  p ) } )  i^i  ( `' G " { ( 2nd `  p ) } ) ) )
108, 9eqtr3i 2465 . . 3  |-  U_ p  e.  ( `' R " D ) ( ( `' F " { ( 1st `  p ) } )  i^i  ( `' G " { ( 2nd `  p ) } ) )  =  ( U_ p  e.  ( ( `' R " D )  i^i  ( ran  F  X.  ran  G
) ) ( ( `' F " { ( 1st `  p ) } )  i^i  ( `' G " { ( 2nd `  p ) } ) )  u. 
U_ p  e.  ( ( `' R " D )  \  ( ran  F  X.  ran  G
) ) ( ( `' F " { ( 1st `  p ) } )  i^i  ( `' G " { ( 2nd `  p ) } ) ) )
115, 10syl6eq 2491 . 2  |-  ( ph  ->  ( `' ( F  oF R G ) " D )  =  ( U_ p  e.  ( ( `' R " D )  i^i  ( ran  F  X.  ran  G
) ) ( ( `' F " { ( 1st `  p ) } )  i^i  ( `' G " { ( 2nd `  p ) } ) )  u. 
U_ p  e.  ( ( `' R " D )  \  ( ran  F  X.  ran  G
) ) ( ( `' F " { ( 1st `  p ) } )  i^i  ( `' G " { ( 2nd `  p ) } ) ) ) )
12 simpr 461 . . . . . . . . . . . 12  |-  ( (
ph  /\  p  e.  ( ( `' R " D )  \  ( ran  F  X.  ran  G
) ) )  ->  p  e.  ( ( `' R " D ) 
\  ( ran  F  X.  ran  G ) ) )
1312eldifbd 3360 . . . . . . . . . . 11  |-  ( (
ph  /\  p  e.  ( ( `' R " D )  \  ( ran  F  X.  ran  G
) ) )  ->  -.  p  e.  ( ran  F  X.  ran  G
) )
14 difssd 3503 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( ( `' R " D )  \  ( ran  F  X.  ran  G
) )  C_  ( `' R " D ) )
15 cnvimass 5208 . . . . . . . . . . . . . . . 16  |-  ( `' R " D ) 
C_  dom  R
16 fndm 5529 . . . . . . . . . . . . . . . . 17  |-  ( R  Fn  ( B  X.  C )  ->  dom  R  =  ( B  X.  C ) )
174, 16syl 16 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  dom  R  =  ( B  X.  C ) )
1815, 17syl5sseq 3423 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( `' R " D )  C_  ( B  X.  C ) )
1914, 18sstrd 3385 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ( `' R " D )  \  ( ran  F  X.  ran  G
) )  C_  ( B  X.  C ) )
2019sselda 3375 . . . . . . . . . . . . 13  |-  ( (
ph  /\  p  e.  ( ( `' R " D )  \  ( ran  F  X.  ran  G
) ) )  ->  p  e.  ( B  X.  C ) )
21 1st2nd2 6632 . . . . . . . . . . . . 13  |-  ( p  e.  ( B  X.  C )  ->  p  =  <. ( 1st `  p
) ,  ( 2nd `  p ) >. )
22 elxp6 6627 . . . . . . . . . . . . . . 15  |-  ( p  e.  ( ran  F  X.  ran  G )  <->  ( p  =  <. ( 1st `  p
) ,  ( 2nd `  p ) >.  /\  (
( 1st `  p
)  e.  ran  F  /\  ( 2nd `  p
)  e.  ran  G
) ) )
2322biimpri 206 . . . . . . . . . . . . . 14  |-  ( ( p  =  <. ( 1st `  p ) ,  ( 2nd `  p
) >.  /\  ( ( 1st `  p )  e. 
ran  F  /\  ( 2nd `  p )  e. 
ran  G ) )  ->  p  e.  ( ran  F  X.  ran  G ) )
2423ex 434 . . . . . . . . . . . . 13  |-  ( p  =  <. ( 1st `  p
) ,  ( 2nd `  p ) >.  ->  (
( ( 1st `  p
)  e.  ran  F  /\  ( 2nd `  p
)  e.  ran  G
)  ->  p  e.  ( ran  F  X.  ran  G ) ) )
2520, 21, 243syl 20 . . . . . . . . . . . 12  |-  ( (
ph  /\  p  e.  ( ( `' R " D )  \  ( ran  F  X.  ran  G
) ) )  -> 
( ( ( 1st `  p )  e.  ran  F  /\  ( 2nd `  p
)  e.  ran  G
)  ->  p  e.  ( ran  F  X.  ran  G ) ) )
2625con3d 133 . . . . . . . . . . 11  |-  ( (
ph  /\  p  e.  ( ( `' R " D )  \  ( ran  F  X.  ran  G
) ) )  -> 
( -.  p  e.  ( ran  F  X.  ran  G )  ->  -.  ( ( 1st `  p
)  e.  ran  F  /\  ( 2nd `  p
)  e.  ran  G
) ) )
2713, 26mpd 15 . . . . . . . . . 10  |-  ( (
ph  /\  p  e.  ( ( `' R " D )  \  ( ran  F  X.  ran  G
) ) )  ->  -.  ( ( 1st `  p
)  e.  ran  F  /\  ( 2nd `  p
)  e.  ran  G
) )
28 ianor 488 . . . . . . . . . 10  |-  ( -.  ( ( 1st `  p
)  e.  ran  F  /\  ( 2nd `  p
)  e.  ran  G
)  <->  ( -.  ( 1st `  p )  e. 
ran  F  \/  -.  ( 2nd `  p )  e.  ran  G ) )
2927, 28sylib 196 . . . . . . . . 9  |-  ( (
ph  /\  p  e.  ( ( `' R " D )  \  ( ran  F  X.  ran  G
) ) )  -> 
( -.  ( 1st `  p )  e.  ran  F  \/  -.  ( 2nd `  p )  e.  ran  G ) )
30 disjsn 3955 . . . . . . . . . 10  |-  ( ( ran  F  i^i  {
( 1st `  p
) } )  =  (/) 
<->  -.  ( 1st `  p
)  e.  ran  F
)
31 disjsn 3955 . . . . . . . . . 10  |-  ( ( ran  G  i^i  {
( 2nd `  p
) } )  =  (/) 
<->  -.  ( 2nd `  p
)  e.  ran  G
)
3230, 31orbi12i 521 . . . . . . . . 9  |-  ( ( ( ran  F  i^i  { ( 1st `  p
) } )  =  (/)  \/  ( ran  G  i^i  { ( 2nd `  p
) } )  =  (/) )  <->  ( -.  ( 1st `  p )  e. 
ran  F  \/  -.  ( 2nd `  p )  e.  ran  G ) )
3329, 32sylibr 212 . . . . . . . 8  |-  ( (
ph  /\  p  e.  ( ( `' R " D )  \  ( ran  F  X.  ran  G
) ) )  -> 
( ( ran  F  i^i  { ( 1st `  p
) } )  =  (/)  \/  ( ran  G  i^i  { ( 2nd `  p
) } )  =  (/) ) )
34 ffn 5578 . . . . . . . . . . . 12  |-  ( F : A --> B  ->  F  Fn  A )
351, 34syl 16 . . . . . . . . . . 11  |-  ( ph  ->  F  Fn  A )
36 dffn3 5585 . . . . . . . . . . 11  |-  ( F  Fn  A  <->  F : A
--> ran  F )
3735, 36sylib 196 . . . . . . . . . 10  |-  ( ph  ->  F : A --> ran  F
)
3837adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  p  e.  ( ( `' R " D )  \  ( ran  F  X.  ran  G
) ) )  ->  F : A --> ran  F
)
39 ffn 5578 . . . . . . . . . . . 12  |-  ( G : A --> C  ->  G  Fn  A )
402, 39syl 16 . . . . . . . . . . 11  |-  ( ph  ->  G  Fn  A )
41 dffn3 5585 . . . . . . . . . . 11  |-  ( G  Fn  A  <->  G : A
--> ran  G )
4240, 41sylib 196 . . . . . . . . . 10  |-  ( ph  ->  G : A --> ran  G
)
4342adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  p  e.  ( ( `' R " D )  \  ( ran  F  X.  ran  G
) ) )  ->  G : A --> ran  G
)
44 fimacnvdisj 5608 . . . . . . . . . . . 12  |-  ( ( F : A --> ran  F  /\  ( ran  F  i^i  { ( 1st `  p
) } )  =  (/) )  ->  ( `' F " { ( 1st `  p ) } )  =  (/) )
45 ineq1 3564 . . . . . . . . . . . . 13  |-  ( ( `' F " { ( 1st `  p ) } )  =  (/)  ->  ( ( `' F " { ( 1st `  p
) } )  i^i  ( `' G " { ( 2nd `  p
) } ) )  =  ( (/)  i^i  ( `' G " { ( 2nd `  p ) } ) ) )
46 incom 3562 . . . . . . . . . . . . . 14  |-  ( (/)  i^i  ( `' G " { ( 2nd `  p
) } ) )  =  ( ( `' G " { ( 2nd `  p ) } )  i^i  (/) )
47 in0 3682 . . . . . . . . . . . . . 14  |-  ( ( `' G " { ( 2nd `  p ) } )  i^i  (/) )  =  (/)
4846, 47eqtri 2463 . . . . . . . . . . . . 13  |-  ( (/)  i^i  ( `' G " { ( 2nd `  p
) } ) )  =  (/)
4945, 48syl6eq 2491 . . . . . . . . . . . 12  |-  ( ( `' F " { ( 1st `  p ) } )  =  (/)  ->  ( ( `' F " { ( 1st `  p
) } )  i^i  ( `' G " { ( 2nd `  p
) } ) )  =  (/) )
5044, 49syl 16 . . . . . . . . . . 11  |-  ( ( F : A --> ran  F  /\  ( ran  F  i^i  { ( 1st `  p
) } )  =  (/) )  ->  ( ( `' F " { ( 1st `  p ) } )  i^i  ( `' G " { ( 2nd `  p ) } ) )  =  (/) )
5150ex 434 . . . . . . . . . 10  |-  ( F : A --> ran  F  ->  ( ( ran  F  i^i  { ( 1st `  p
) } )  =  (/)  ->  ( ( `' F " { ( 1st `  p ) } )  i^i  ( `' G " { ( 2nd `  p ) } ) )  =  (/) ) )
52 fimacnvdisj 5608 . . . . . . . . . . . 12  |-  ( ( G : A --> ran  G  /\  ( ran  G  i^i  { ( 2nd `  p
) } )  =  (/) )  ->  ( `' G " { ( 2nd `  p ) } )  =  (/) )
53 ineq2 3565 . . . . . . . . . . . . 13  |-  ( ( `' G " { ( 2nd `  p ) } )  =  (/)  ->  ( ( `' F " { ( 1st `  p
) } )  i^i  ( `' G " { ( 2nd `  p
) } ) )  =  ( ( `' F " { ( 1st `  p ) } )  i^i  (/) ) )
54 in0 3682 . . . . . . . . . . . . 13  |-  ( ( `' F " { ( 1st `  p ) } )  i^i  (/) )  =  (/)
5553, 54syl6eq 2491 . . . . . . . . . . . 12  |-  ( ( `' G " { ( 2nd `  p ) } )  =  (/)  ->  ( ( `' F " { ( 1st `  p
) } )  i^i  ( `' G " { ( 2nd `  p
) } ) )  =  (/) )
5652, 55syl 16 . . . . . . . . . . 11  |-  ( ( G : A --> ran  G  /\  ( ran  G  i^i  { ( 2nd `  p
) } )  =  (/) )  ->  ( ( `' F " { ( 1st `  p ) } )  i^i  ( `' G " { ( 2nd `  p ) } ) )  =  (/) )
5756ex 434 . . . . . . . . . 10  |-  ( G : A --> ran  G  ->  ( ( ran  G  i^i  { ( 2nd `  p
) } )  =  (/)  ->  ( ( `' F " { ( 1st `  p ) } )  i^i  ( `' G " { ( 2nd `  p ) } ) )  =  (/) ) )
5851, 57jaao 509 . . . . . . . . 9  |-  ( ( F : A --> ran  F  /\  G : A --> ran  G
)  ->  ( (
( ran  F  i^i  { ( 1st `  p
) } )  =  (/)  \/  ( ran  G  i^i  { ( 2nd `  p
) } )  =  (/) )  ->  ( ( `' F " { ( 1st `  p ) } )  i^i  ( `' G " { ( 2nd `  p ) } ) )  =  (/) ) )
5938, 43, 58syl2anc 661 . . . . . . . 8  |-  ( (
ph  /\  p  e.  ( ( `' R " D )  \  ( ran  F  X.  ran  G
) ) )  -> 
( ( ( ran 
F  i^i  { ( 1st `  p ) } )  =  (/)  \/  ( ran  G  i^i  { ( 2nd `  p ) } )  =  (/) )  ->  ( ( `' F " { ( 1st `  p ) } )  i^i  ( `' G " { ( 2nd `  p ) } ) )  =  (/) ) )
6033, 59mpd 15 . . . . . . 7  |-  ( (
ph  /\  p  e.  ( ( `' R " D )  \  ( ran  F  X.  ran  G
) ) )  -> 
( ( `' F " { ( 1st `  p
) } )  i^i  ( `' G " { ( 2nd `  p
) } ) )  =  (/) )
6160ralrimiva 2818 . . . . . 6  |-  ( ph  ->  A. p  e.  ( ( `' R " D )  \  ( ran  F  X.  ran  G
) ) ( ( `' F " { ( 1st `  p ) } )  i^i  ( `' G " { ( 2nd `  p ) } ) )  =  (/) )
62 iuneq2 4206 . . . . . 6  |-  ( A. p  e.  ( ( `' R " D ) 
\  ( ran  F  X.  ran  G ) ) ( ( `' F " { ( 1st `  p
) } )  i^i  ( `' G " { ( 2nd `  p
) } ) )  =  (/)  ->  U_ p  e.  ( ( `' R " D )  \  ( ran  F  X.  ran  G
) ) ( ( `' F " { ( 1st `  p ) } )  i^i  ( `' G " { ( 2nd `  p ) } ) )  = 
U_ p  e.  ( ( `' R " D )  \  ( ran  F  X.  ran  G
) ) (/) )
6361, 62syl 16 . . . . 5  |-  ( ph  ->  U_ p  e.  ( ( `' R " D )  \  ( ran  F  X.  ran  G
) ) ( ( `' F " { ( 1st `  p ) } )  i^i  ( `' G " { ( 2nd `  p ) } ) )  = 
U_ p  e.  ( ( `' R " D )  \  ( ran  F  X.  ran  G
) ) (/) )
64 iun0 4245 . . . . 5  |-  U_ p  e.  ( ( `' R " D )  \  ( ran  F  X.  ran  G
) ) (/)  =  (/)
6563, 64syl6eq 2491 . . . 4  |-  ( ph  ->  U_ p  e.  ( ( `' R " D )  \  ( ran  F  X.  ran  G
) ) ( ( `' F " { ( 1st `  p ) } )  i^i  ( `' G " { ( 2nd `  p ) } ) )  =  (/) )
6665uneq2d 3529 . . 3  |-  ( ph  ->  ( U_ p  e.  ( ( `' R " D )  i^i  ( ran  F  X.  ran  G
) ) ( ( `' F " { ( 1st `  p ) } )  i^i  ( `' G " { ( 2nd `  p ) } ) )  u. 
U_ p  e.  ( ( `' R " D )  \  ( ran  F  X.  ran  G
) ) ( ( `' F " { ( 1st `  p ) } )  i^i  ( `' G " { ( 2nd `  p ) } ) ) )  =  ( U_ p  e.  ( ( `' R " D )  i^i  ( ran  F  X.  ran  G
) ) ( ( `' F " { ( 1st `  p ) } )  i^i  ( `' G " { ( 2nd `  p ) } ) )  u.  (/) ) )
67 un0 3681 . . 3  |-  ( U_ p  e.  ( ( `' R " D )  i^i  ( ran  F  X.  ran  G ) ) ( ( `' F " { ( 1st `  p
) } )  i^i  ( `' G " { ( 2nd `  p
) } ) )  u.  (/) )  =  U_ p  e.  ( ( `' R " D )  i^i  ( ran  F  X.  ran  G ) ) ( ( `' F " { ( 1st `  p
) } )  i^i  ( `' G " { ( 2nd `  p
) } ) )
6866, 67syl6eq 2491 . 2  |-  ( ph  ->  ( U_ p  e.  ( ( `' R " D )  i^i  ( ran  F  X.  ran  G
) ) ( ( `' F " { ( 1st `  p ) } )  i^i  ( `' G " { ( 2nd `  p ) } ) )  u. 
U_ p  e.  ( ( `' R " D )  \  ( ran  F  X.  ran  G
) ) ( ( `' F " { ( 1st `  p ) } )  i^i  ( `' G " { ( 2nd `  p ) } ) ) )  =  U_ p  e.  ( ( `' R " D )  i^i  ( ran  F  X.  ran  G
) ) ( ( `' F " { ( 1st `  p ) } )  i^i  ( `' G " { ( 2nd `  p ) } ) ) )
6911, 68eqtrd 2475 1  |-  ( ph  ->  ( `' ( F  oF R G ) " D )  =  U_ p  e.  ( ( `' R " D )  i^i  ( ran  F  X.  ran  G
) ) ( ( `' F " { ( 1st `  p ) } )  i^i  ( `' G " { ( 2nd `  p ) } ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 368    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2734    \ cdif 3344    u. cun 3345    i^i cin 3346   (/)c0 3656   {csn 3896   <.cop 3902   U_ciun 4190    X. cxp 4857   `'ccnv 4858   dom cdm 4859   ran crn 4860   "cima 4862    Fn wfn 5432   -->wf 5433   ` cfv 5437  (class class class)co 6110    oFcof 6337   1stc1st 6594   2ndc2nd 6595
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 4422  ax-sep 4432  ax-nul 4440  ax-pow 4489  ax-pr 4550  ax-un 6391
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 2577  df-ne 2622  df-ral 2739  df-rex 2740  df-reu 2741  df-rab 2743  df-v 2993  df-sbc 3206  df-csb 3308  df-dif 3350  df-un 3352  df-in 3354  df-ss 3361  df-nul 3657  df-if 3811  df-sn 3897  df-pr 3899  df-op 3903  df-uni 4111  df-iun 4192  df-br 4312  df-opab 4370  df-mpt 4371  df-id 4655  df-xp 4865  df-rel 4866  df-cnv 4867  df-co 4868  df-dm 4869  df-rn 4870  df-res 4871  df-ima 4872  df-iota 5400  df-fun 5439  df-fn 5440  df-f 5441  df-f1 5442  df-fo 5443  df-f1o 5444  df-fv 5445  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-of 6339  df-1st 6596  df-2nd 6597
This theorem is referenced by:  sibfof  26745
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