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Theorem ofpreima2 27735
Description: Express the preimage of a function operation as a union of preimages. This version of ofpreima 27734 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 27734 . . 3  |-  ( ph  ->  ( `' ( F  oF R G ) " D )  =  U_ p  e.  ( `' R " D ) ( ( `' F " { ( 1st `  p ) } )  i^i  ( `' G " { ( 2nd `  p ) } ) ) )
6 inundif 3894 . . . . 5  |-  ( ( ( `' R " D )  i^i  ( ran  F  X.  ran  G
) )  u.  (
( `' R " D )  \  ( ran  F  X.  ran  G
) ) )  =  ( `' R " D )
7 iuneq1 4329 . . . . 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 4400 . . . 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 2485 . . 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 2511 . 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 459 . . . . . . . . . . 11  |-  ( (
ph  /\  p  e.  ( ( `' R " D )  \  ( ran  F  X.  ran  G
) ) )  ->  p  e.  ( ( `' R " D ) 
\  ( ran  F  X.  ran  G ) ) )
1312eldifbd 3474 . . . . . . . . . 10  |-  ( (
ph  /\  p  e.  ( ( `' R " D )  \  ( ran  F  X.  ran  G
) ) )  ->  -.  p  e.  ( ran  F  X.  ran  G
) )
14 cnvimass 5345 . . . . . . . . . . . . . 14  |-  ( `' R " D ) 
C_  dom  R
15 fndm 5662 . . . . . . . . . . . . . . 15  |-  ( R  Fn  ( B  X.  C )  ->  dom  R  =  ( B  X.  C ) )
164, 15syl 16 . . . . . . . . . . . . . 14  |-  ( ph  ->  dom  R  =  ( B  X.  C ) )
1714, 16syl5sseq 3537 . . . . . . . . . . . . 13  |-  ( ph  ->  ( `' R " D )  C_  ( B  X.  C ) )
1817ssdifssd 3628 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( `' R " D )  \  ( ran  F  X.  ran  G
) )  C_  ( B  X.  C ) )
1918sselda 3489 . . . . . . . . . . 11  |-  ( (
ph  /\  p  e.  ( ( `' R " D )  \  ( ran  F  X.  ran  G
) ) )  ->  p  e.  ( B  X.  C ) )
20 1st2nd2 6810 . . . . . . . . . . 11  |-  ( p  e.  ( B  X.  C )  ->  p  =  <. ( 1st `  p
) ,  ( 2nd `  p ) >. )
21 elxp6 6805 . . . . . . . . . . . 12  |-  ( p  e.  ( ran  F  X.  ran  G )  <->  ( p  =  <. ( 1st `  p
) ,  ( 2nd `  p ) >.  /\  (
( 1st `  p
)  e.  ran  F  /\  ( 2nd `  p
)  e.  ran  G
) ) )
2221simplbi2 623 . . . . . . . . . . 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 20 . . . . . . . . . 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 177 . . . . . . . . 9  |-  ( (
ph  /\  p  e.  ( ( `' R " D )  \  ( ran  F  X.  ran  G
) ) )  ->  -.  ( ( 1st `  p
)  e.  ran  F  /\  ( 2nd `  p
)  e.  ran  G
) )
25 ianor 486 . . . . . . . . 9  |-  ( -.  ( ( 1st `  p
)  e.  ran  F  /\  ( 2nd `  p
)  e.  ran  G
)  <->  ( -.  ( 1st `  p )  e. 
ran  F  \/  -.  ( 2nd `  p )  e.  ran  G ) )
2624, 25sylib 196 . . . . . . . 8  |-  ( (
ph  /\  p  e.  ( ( `' R " D )  \  ( ran  F  X.  ran  G
) ) )  -> 
( -.  ( 1st `  p )  e.  ran  F  \/  -.  ( 2nd `  p )  e.  ran  G ) )
27 disjsn 4076 . . . . . . . . 9  |-  ( ( ran  F  i^i  {
( 1st `  p
) } )  =  (/) 
<->  -.  ( 1st `  p
)  e.  ran  F
)
28 disjsn 4076 . . . . . . . . 9  |-  ( ( ran  G  i^i  {
( 2nd `  p
) } )  =  (/) 
<->  -.  ( 2nd `  p
)  e.  ran  G
)
2927, 28orbi12i 519 . . . . . . . 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 212 . . . . . . 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 5713 . . . . . . . . . . 11  |-  ( F : A --> B  ->  F  Fn  A )
321, 31syl 16 . . . . . . . . . 10  |-  ( ph  ->  F  Fn  A )
33 dffn3 5720 . . . . . . . . . 10  |-  ( F  Fn  A  <->  F : A
--> ran  F )
3432, 33sylib 196 . . . . . . . . 9  |-  ( ph  ->  F : A --> ran  F
)
3534adantr 463 . . . . . . . 8  |-  ( (
ph  /\  p  e.  ( ( `' R " D )  \  ( ran  F  X.  ran  G
) ) )  ->  F : A --> ran  F
)
36 ffn 5713 . . . . . . . . . . 11  |-  ( G : A --> C  ->  G  Fn  A )
372, 36syl 16 . . . . . . . . . 10  |-  ( ph  ->  G  Fn  A )
38 dffn3 5720 . . . . . . . . . 10  |-  ( G  Fn  A  <->  G : A
--> ran  G )
3937, 38sylib 196 . . . . . . . . 9  |-  ( ph  ->  G : A --> ran  G
)
4039adantr 463 . . . . . . . 8  |-  ( (
ph  /\  p  e.  ( ( `' R " D )  \  ( ran  F  X.  ran  G
) ) )  ->  G : A --> ran  G
)
41 fimacnvdisj 5745 . . . . . . . . . . 11  |-  ( ( F : A --> ran  F  /\  ( ran  F  i^i  { ( 1st `  p
) } )  =  (/) )  ->  ( `' F " { ( 1st `  p ) } )  =  (/) )
42 ineq1 3679 . . . . . . . . . . . 12  |-  ( ( `' F " { ( 1st `  p ) } )  =  (/)  ->  ( ( `' F " { ( 1st `  p
) } )  i^i  ( `' G " { ( 2nd `  p
) } ) )  =  ( (/)  i^i  ( `' G " { ( 2nd `  p ) } ) ) )
43 incom 3677 . . . . . . . . . . . . 13  |-  ( (/)  i^i  ( `' G " { ( 2nd `  p
) } ) )  =  ( ( `' G " { ( 2nd `  p ) } )  i^i  (/) )
44 in0 3810 . . . . . . . . . . . . 13  |-  ( ( `' G " { ( 2nd `  p ) } )  i^i  (/) )  =  (/)
4543, 44eqtri 2483 . . . . . . . . . . . 12  |-  ( (/)  i^i  ( `' G " { ( 2nd `  p
) } ) )  =  (/)
4642, 45syl6eq 2511 . . . . . . . . . . 11  |-  ( ( `' F " { ( 1st `  p ) } )  =  (/)  ->  ( ( `' F " { ( 1st `  p
) } )  i^i  ( `' G " { ( 2nd `  p
) } ) )  =  (/) )
4741, 46syl 16 . . . . . . . . . 10  |-  ( ( F : A --> ran  F  /\  ( ran  F  i^i  { ( 1st `  p
) } )  =  (/) )  ->  ( ( `' F " { ( 1st `  p ) } )  i^i  ( `' G " { ( 2nd `  p ) } ) )  =  (/) )
4847ex 432 . . . . . . . . 9  |-  ( F : A --> ran  F  ->  ( ( ran  F  i^i  { ( 1st `  p
) } )  =  (/)  ->  ( ( `' F " { ( 1st `  p ) } )  i^i  ( `' G " { ( 2nd `  p ) } ) )  =  (/) ) )
49 fimacnvdisj 5745 . . . . . . . . . . 11  |-  ( ( G : A --> ran  G  /\  ( ran  G  i^i  { ( 2nd `  p
) } )  =  (/) )  ->  ( `' G " { ( 2nd `  p ) } )  =  (/) )
50 ineq2 3680 . . . . . . . . . . . 12  |-  ( ( `' G " { ( 2nd `  p ) } )  =  (/)  ->  ( ( `' F " { ( 1st `  p
) } )  i^i  ( `' G " { ( 2nd `  p
) } ) )  =  ( ( `' F " { ( 1st `  p ) } )  i^i  (/) ) )
51 in0 3810 . . . . . . . . . . . 12  |-  ( ( `' F " { ( 1st `  p ) } )  i^i  (/) )  =  (/)
5250, 51syl6eq 2511 . . . . . . . . . . 11  |-  ( ( `' G " { ( 2nd `  p ) } )  =  (/)  ->  ( ( `' F " { ( 1st `  p
) } )  i^i  ( `' G " { ( 2nd `  p
) } ) )  =  (/) )
5349, 52syl 16 . . . . . . . . . 10  |-  ( ( G : A --> ran  G  /\  ( ran  G  i^i  { ( 2nd `  p
) } )  =  (/) )  ->  ( ( `' F " { ( 1st `  p ) } )  i^i  ( `' G " { ( 2nd `  p ) } ) )  =  (/) )
5453ex 432 . . . . . . . . 9  |-  ( G : A --> ran  G  ->  ( ( ran  G  i^i  { ( 2nd `  p
) } )  =  (/)  ->  ( ( `' F " { ( 1st `  p ) } )  i^i  ( `' G " { ( 2nd `  p ) } ) )  =  (/) ) )
5548, 54jaao 507 . . . . . . . 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 659 . . . . . . 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 4337 . . . . 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 4371 . . . . 5  |-  U_ p  e.  ( ( `' R " D )  \  ( ran  F  X.  ran  G
) ) (/)  =  (/)
6058, 59syl6eq 2511 . . . 4  |-  ( ph  ->  U_ p  e.  ( ( `' R " D )  \  ( ran  F  X.  ran  G
) ) ( ( `' F " { ( 1st `  p ) } )  i^i  ( `' G " { ( 2nd `  p ) } ) )  =  (/) )
6160uneq2d 3644 . . 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 3809 . . 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 2511 . 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 2495 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 366    /\ wa 367    = wceq 1398    e. wcel 1823    \ cdif 3458    u. cun 3459    i^i cin 3460   (/)c0 3783   {csn 4016   <.cop 4022   U_ciun 4315    X. cxp 4986   `'ccnv 4987   dom cdm 4988   ran crn 4989   "cima 4991    Fn wfn 5565   -->wf 5566   ` cfv 5570  (class class class)co 6270    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:  sibfof  28546
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