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Theorem ofpreima2 28256
Description: Express the preimage of a function operation as a union of preimages. This version of ofpreima 28255 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 28255 . . 3  |-  ( ph  ->  ( `' ( F  oF R G ) " D )  =  U_ p  e.  ( `' R " D ) ( ( `' F " { ( 1st `  p ) } )  i^i  ( `' G " { ( 2nd `  p ) } ) ) )
6 inundif 3873 . . . . 5  |-  ( ( ( `' R " D )  i^i  ( ran  F  X.  ran  G
) )  u.  (
( `' R " D )  \  ( ran  F  X.  ran  G
) ) )  =  ( `' R " D )
7 iuneq1 4310 . . . . 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 4381 . . . 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 2453 . . 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 2479 . 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 462 . . . . . . . . . . 11  |-  ( (
ph  /\  p  e.  ( ( `' R " D )  \  ( ran  F  X.  ran  G
) ) )  ->  p  e.  ( ( `' R " D ) 
\  ( ran  F  X.  ran  G ) ) )
1312eldifbd 3449 . . . . . . . . . 10  |-  ( (
ph  /\  p  e.  ( ( `' R " D )  \  ( ran  F  X.  ran  G
) ) )  ->  -.  p  e.  ( ran  F  X.  ran  G
) )
14 cnvimass 5203 . . . . . . . . . . . . . 14  |-  ( `' R " D ) 
C_  dom  R
15 fndm 5689 . . . . . . . . . . . . . . 15  |-  ( R  Fn  ( B  X.  C )  ->  dom  R  =  ( B  X.  C ) )
164, 15syl 17 . . . . . . . . . . . . . 14  |-  ( ph  ->  dom  R  =  ( B  X.  C ) )
1714, 16syl5sseq 3512 . . . . . . . . . . . . 13  |-  ( ph  ->  ( `' R " D )  C_  ( B  X.  C ) )
1817ssdifssd 3603 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( `' R " D )  \  ( ran  F  X.  ran  G
) )  C_  ( B  X.  C ) )
1918sselda 3464 . . . . . . . . . . 11  |-  ( (
ph  /\  p  e.  ( ( `' R " D )  \  ( ran  F  X.  ran  G
) ) )  ->  p  e.  ( B  X.  C ) )
20 1st2nd2 6840 . . . . . . . . . . 11  |-  ( p  e.  ( B  X.  C )  ->  p  =  <. ( 1st `  p
) ,  ( 2nd `  p ) >. )
21 elxp6 6835 . . . . . . . . . . . 12  |-  ( p  e.  ( ran  F  X.  ran  G )  <->  ( p  =  <. ( 1st `  p
) ,  ( 2nd `  p ) >.  /\  (
( 1st `  p
)  e.  ran  F  /\  ( 2nd `  p
)  e.  ran  G
) ) )
2221simplbi2 629 . . . . . . . . . . 11  |-  ( p  =  <. ( 1st `  p
) ,  ( 2nd `  p ) >.  ->  (
( ( 1st `  p
)  e.  ran  F  /\  ( 2nd `  p
)  e.  ran  G
)  ->  p  e.  ( ran  F  X.  ran  G ) ) )
2319, 20, 223syl 18 . . . . . . . . . 10  |-  ( (
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 ) ) )
2413, 23mtod 180 . . . . . . . . 9  |-  ( (
ph  /\  p  e.  ( ( `' R " D )  \  ( ran  F  X.  ran  G
) ) )  ->  -.  ( ( 1st `  p
)  e.  ran  F  /\  ( 2nd `  p
)  e.  ran  G
) )
25 ianor 490 . . . . . . . . 9  |-  ( -.  ( ( 1st `  p
)  e.  ran  F  /\  ( 2nd `  p
)  e.  ran  G
)  <->  ( -.  ( 1st `  p )  e. 
ran  F  \/  -.  ( 2nd `  p )  e.  ran  G ) )
2624, 25sylib 199 . . . . . . . 8  |-  ( (
ph  /\  p  e.  ( ( `' R " D )  \  ( ran  F  X.  ran  G
) ) )  -> 
( -.  ( 1st `  p )  e.  ran  F  \/  -.  ( 2nd `  p )  e.  ran  G ) )
27 disjsn 4057 . . . . . . . . 9  |-  ( ( ran  F  i^i  {
( 1st `  p
) } )  =  (/) 
<->  -.  ( 1st `  p
)  e.  ran  F
)
28 disjsn 4057 . . . . . . . . 9  |-  ( ( ran  G  i^i  {
( 2nd `  p
) } )  =  (/) 
<->  -.  ( 2nd `  p
)  e.  ran  G
)
2927, 28orbi12i 523 . . . . . . . 8  |-  ( ( ( ran  F  i^i  { ( 1st `  p
) } )  =  (/)  \/  ( ran  G  i^i  { ( 2nd `  p
) } )  =  (/) )  <->  ( -.  ( 1st `  p )  e. 
ran  F  \/  -.  ( 2nd `  p )  e.  ran  G ) )
3026, 29sylibr 215 . . . . . . 7  |-  ( (
ph  /\  p  e.  ( ( `' R " D )  \  ( ran  F  X.  ran  G
) ) )  -> 
( ( ran  F  i^i  { ( 1st `  p
) } )  =  (/)  \/  ( ran  G  i^i  { ( 2nd `  p
) } )  =  (/) ) )
31 ffn 5742 . . . . . . . . . . 11  |-  ( F : A --> B  ->  F  Fn  A )
321, 31syl 17 . . . . . . . . . 10  |-  ( ph  ->  F  Fn  A )
33 dffn3 5749 . . . . . . . . . 10  |-  ( F  Fn  A  <->  F : A
--> ran  F )
3432, 33sylib 199 . . . . . . . . 9  |-  ( ph  ->  F : A --> ran  F
)
3534adantr 466 . . . . . . . 8  |-  ( (
ph  /\  p  e.  ( ( `' R " D )  \  ( ran  F  X.  ran  G
) ) )  ->  F : A --> ran  F
)
36 ffn 5742 . . . . . . . . . . 11  |-  ( G : A --> C  ->  G  Fn  A )
372, 36syl 17 . . . . . . . . . 10  |-  ( ph  ->  G  Fn  A )
38 dffn3 5749 . . . . . . . . . 10  |-  ( G  Fn  A  <->  G : A
--> ran  G )
3937, 38sylib 199 . . . . . . . . 9  |-  ( ph  ->  G : A --> ran  G
)
4039adantr 466 . . . . . . . 8  |-  ( (
ph  /\  p  e.  ( ( `' R " D )  \  ( ran  F  X.  ran  G
) ) )  ->  G : A --> ran  G
)
41 fimacnvdisj 5774 . . . . . . . . . . 11  |-  ( ( F : A --> ran  F  /\  ( ran  F  i^i  { ( 1st `  p
) } )  =  (/) )  ->  ( `' F " { ( 1st `  p ) } )  =  (/) )
42 ineq1 3657 . . . . . . . . . . . 12  |-  ( ( `' F " { ( 1st `  p ) } )  =  (/)  ->  ( ( `' F " { ( 1st `  p
) } )  i^i  ( `' G " { ( 2nd `  p
) } ) )  =  ( (/)  i^i  ( `' G " { ( 2nd `  p ) } ) ) )
43 incom 3655 . . . . . . . . . . . . 13  |-  ( (/)  i^i  ( `' G " { ( 2nd `  p
) } ) )  =  ( ( `' G " { ( 2nd `  p ) } )  i^i  (/) )
44 in0 3788 . . . . . . . . . . . . 13  |-  ( ( `' G " { ( 2nd `  p ) } )  i^i  (/) )  =  (/)
4543, 44eqtri 2451 . . . . . . . . . . . 12  |-  ( (/)  i^i  ( `' G " { ( 2nd `  p
) } ) )  =  (/)
4642, 45syl6eq 2479 . . . . . . . . . . 11  |-  ( ( `' F " { ( 1st `  p ) } )  =  (/)  ->  ( ( `' F " { ( 1st `  p
) } )  i^i  ( `' G " { ( 2nd `  p
) } ) )  =  (/) )
4741, 46syl 17 . . . . . . . . . 10  |-  ( ( F : A --> ran  F  /\  ( ran  F  i^i  { ( 1st `  p
) } )  =  (/) )  ->  ( ( `' F " { ( 1st `  p ) } )  i^i  ( `' G " { ( 2nd `  p ) } ) )  =  (/) )
4847ex 435 . . . . . . . . 9  |-  ( F : A --> ran  F  ->  ( ( ran  F  i^i  { ( 1st `  p
) } )  =  (/)  ->  ( ( `' F " { ( 1st `  p ) } )  i^i  ( `' G " { ( 2nd `  p ) } ) )  =  (/) ) )
49 fimacnvdisj 5774 . . . . . . . . . . 11  |-  ( ( G : A --> ran  G  /\  ( ran  G  i^i  { ( 2nd `  p
) } )  =  (/) )  ->  ( `' G " { ( 2nd `  p ) } )  =  (/) )
50 ineq2 3658 . . . . . . . . . . . 12  |-  ( ( `' G " { ( 2nd `  p ) } )  =  (/)  ->  ( ( `' F " { ( 1st `  p
) } )  i^i  ( `' G " { ( 2nd `  p
) } ) )  =  ( ( `' F " { ( 1st `  p ) } )  i^i  (/) ) )
51 in0 3788 . . . . . . . . . . . 12  |-  ( ( `' F " { ( 1st `  p ) } )  i^i  (/) )  =  (/)
5250, 51syl6eq 2479 . . . . . . . . . . 11  |-  ( ( `' G " { ( 2nd `  p ) } )  =  (/)  ->  ( ( `' F " { ( 1st `  p
) } )  i^i  ( `' G " { ( 2nd `  p
) } ) )  =  (/) )
5349, 52syl 17 . . . . . . . . . 10  |-  ( ( G : A --> ran  G  /\  ( ran  G  i^i  { ( 2nd `  p
) } )  =  (/) )  ->  ( ( `' F " { ( 1st `  p ) } )  i^i  ( `' G " { ( 2nd `  p ) } ) )  =  (/) )
5453ex 435 . . . . . . . . 9  |-  ( G : A --> ran  G  ->  ( ( ran  G  i^i  { ( 2nd `  p
) } )  =  (/)  ->  ( ( `' F " { ( 1st `  p ) } )  i^i  ( `' G " { ( 2nd `  p ) } ) )  =  (/) ) )
5548, 54jaao 511 . . . . . . . 8  |-  ( ( 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 ) } ) )  =  (/) ) )
5635, 40, 55syl2anc 665 . . . . . . 7  |-  ( (
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 ) } ) )  =  (/) ) )
5730, 56mpd 15 . . . . . 6  |-  ( (
ph  /\  p  e.  ( ( `' R " D )  \  ( ran  F  X.  ran  G
) ) )  -> 
( ( `' F " { ( 1st `  p
) } )  i^i  ( `' G " { ( 2nd `  p
) } ) )  =  (/) )
5857iuneq2dv 4318 . . . . 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
) ) (/) )
59 iun0 4352 . . . . 5  |-  U_ p  e.  ( ( `' R " D )  \  ( ran  F  X.  ran  G
) ) (/)  =  (/)
6058, 59syl6eq 2479 . . . 4  |-  ( ph  ->  U_ p  e.  ( ( `' R " D )  \  ( ran  F  X.  ran  G
) ) ( ( `' F " { ( 1st `  p ) } )  i^i  ( `' G " { ( 2nd `  p ) } ) )  =  (/) )
6160uneq2d 3620 . . 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.  (/) ) )
62 un0 3787 . . 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
) } ) )
6361, 62syl6eq 2479 . 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 ) } ) ) )
6411, 63eqtrd 2463 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 369    /\ wa 370    = wceq 1437    e. wcel 1868    \ cdif 3433    u. cun 3434    i^i cin 3435   (/)c0 3761   {csn 3996   <.cop 4002   U_ciun 4296    X. cxp 4847   `'ccnv 4848   dom cdm 4849   ran crn 4850   "cima 4852    Fn wfn 5592   -->wf 5593   ` cfv 5597  (class class class)co 6301    oFcof 6539   1stc1st 6801   2ndc2nd 6802
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-rep 4533  ax-sep 4543  ax-nul 4551  ax-pow 4598  ax-pr 4656  ax-un 6593
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-ral 2780  df-rex 2781  df-reu 2782  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3910  df-sn 3997  df-pr 3999  df-op 4003  df-uni 4217  df-iun 4298  df-br 4421  df-opab 4480  df-mpt 4481  df-id 4764  df-xp 4855  df-rel 4856  df-cnv 4857  df-co 4858  df-dm 4859  df-rn 4860  df-res 4861  df-ima 4862  df-iota 5561  df-fun 5599  df-fn 5600  df-f 5601  df-f1 5602  df-fo 5603  df-f1o 5604  df-fv 5605  df-ov 6304  df-oprab 6305  df-mpt2 6306  df-of 6541  df-1st 6803  df-2nd 6804
This theorem is referenced by:  sibfof  29166
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