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Theorem imadifxp 27672
Description: Image of the difference with a Cartesian product (Contributed by Thierry Arnoux, 13-Dec-2017.)
Assertion
Ref Expression
imadifxp  |-  ( C 
C_  A  ->  (
( R  \  ( A  X.  B ) )
" C )  =  ( ( R " C )  \  B
) )

Proof of Theorem imadifxp
StepHypRef Expression
1 ima0 5340 . . . 4  |-  ( ( R  \  ( A  X.  B ) )
" (/) )  =  (/)
2 imaeq2 5321 . . . 4  |-  ( C  =  (/)  ->  ( ( R  \  ( A  X.  B ) )
" C )  =  ( ( R  \ 
( A  X.  B
) ) " (/) ) )
3 imaeq2 5321 . . . . . . 7  |-  ( C  =  (/)  ->  ( R
" C )  =  ( R " (/) ) )
4 ima0 5340 . . . . . . 7  |-  ( R
" (/) )  =  (/)
53, 4syl6eq 2511 . . . . . 6  |-  ( C  =  (/)  ->  ( R
" C )  =  (/) )
65difeq1d 3607 . . . . 5  |-  ( C  =  (/)  ->  ( ( R " C ) 
\  B )  =  ( (/)  \  B ) )
7 0dif 3887 . . . . 5  |-  ( (/)  \  B )  =  (/)
86, 7syl6eq 2511 . . . 4  |-  ( C  =  (/)  ->  ( ( R " C ) 
\  B )  =  (/) )
91, 2, 83eqtr4a 2521 . . 3  |-  ( C  =  (/)  ->  ( ( R  \  ( A  X.  B ) )
" C )  =  ( ( R " C )  \  B
) )
109adantl 464 . 2  |-  ( ( C  C_  A  /\  C  =  (/) )  -> 
( ( R  \ 
( A  X.  B
) ) " C
)  =  ( ( R " C ) 
\  B ) )
11 inundif 3894 . . . . . . . . 9  |-  ( ( R  i^i  ( A  X.  B ) )  u.  ( R  \ 
( A  X.  B
) ) )  =  R
1211imaeq1i 5322 . . . . . . . 8  |-  ( ( ( R  i^i  ( A  X.  B ) )  u.  ( R  \ 
( A  X.  B
) ) ) " C )  =  ( R " C )
13 imaundir 5404 . . . . . . . 8  |-  ( ( ( R  i^i  ( A  X.  B ) )  u.  ( R  \ 
( A  X.  B
) ) ) " C )  =  ( ( ( R  i^i  ( A  X.  B
) ) " C
)  u.  ( ( R  \  ( A  X.  B ) )
" C ) )
1412, 13eqtr3i 2485 . . . . . . 7  |-  ( R
" C )  =  ( ( ( R  i^i  ( A  X.  B ) ) " C )  u.  (
( R  \  ( A  X.  B ) )
" C ) )
1514difeq1i 3604 . . . . . 6  |-  ( ( R " C ) 
\  B )  =  ( ( ( ( R  i^i  ( A  X.  B ) )
" C )  u.  ( ( R  \ 
( A  X.  B
) ) " C
) )  \  B
)
16 difundir 3748 . . . . . 6  |-  ( ( ( ( R  i^i  ( A  X.  B
) ) " C
)  u.  ( ( R  \  ( A  X.  B ) )
" C ) ) 
\  B )  =  ( ( ( ( R  i^i  ( A  X.  B ) )
" C )  \  B )  u.  (
( ( R  \ 
( A  X.  B
) ) " C
)  \  B )
)
1715, 16eqtri 2483 . . . . 5  |-  ( ( R " C ) 
\  B )  =  ( ( ( ( R  i^i  ( A  X.  B ) )
" C )  \  B )  u.  (
( ( R  \ 
( A  X.  B
) ) " C
)  \  B )
)
18 inss2 3705 . . . . . . . . 9  |-  ( R  i^i  ( A  X.  B ) )  C_  ( A  X.  B
)
19 imass1 5359 . . . . . . . . 9  |-  ( ( R  i^i  ( A  X.  B ) ) 
C_  ( A  X.  B )  ->  (
( R  i^i  ( A  X.  B ) )
" C )  C_  ( ( A  X.  B ) " C
) )
20 ssdif 3625 . . . . . . . . 9  |-  ( ( ( R  i^i  ( A  X.  B ) )
" C )  C_  ( ( A  X.  B ) " C
)  ->  ( (
( R  i^i  ( A  X.  B ) )
" C )  \  B )  C_  (
( ( A  X.  B ) " C
)  \  B )
)
2118, 19, 20mp2b 10 . . . . . . . 8  |-  ( ( ( R  i^i  ( A  X.  B ) )
" C )  \  B )  C_  (
( ( A  X.  B ) " C
)  \  B )
22 xpima 5434 . . . . . . . . . . 11  |-  ( ( A  X.  B )
" C )  =  if ( ( A  i^i  C )  =  (/) ,  (/) ,  B )
23 incom 3677 . . . . . . . . . . . . . . 15  |-  ( C  i^i  A )  =  ( A  i^i  C
)
24 df-ss 3475 . . . . . . . . . . . . . . . 16  |-  ( C 
C_  A  <->  ( C  i^i  A )  =  C )
2524biimpi 194 . . . . . . . . . . . . . . 15  |-  ( C 
C_  A  ->  ( C  i^i  A )  =  C )
2623, 25syl5eqr 2509 . . . . . . . . . . . . . 14  |-  ( C 
C_  A  ->  ( A  i^i  C )  =  C )
2726adantl 464 . . . . . . . . . . . . 13  |-  ( ( C  =/=  (/)  /\  C  C_  A )  ->  ( A  i^i  C )  =  C )
28 simpl 455 . . . . . . . . . . . . 13  |-  ( ( C  =/=  (/)  /\  C  C_  A )  ->  C  =/=  (/) )
2927, 28eqnetrd 2747 . . . . . . . . . . . 12  |-  ( ( C  =/=  (/)  /\  C  C_  A )  ->  ( A  i^i  C )  =/=  (/) )
30 df-ne 2651 . . . . . . . . . . . . 13  |-  ( ( A  i^i  C )  =/=  (/)  <->  -.  ( A  i^i  C )  =  (/) )
3130biimpi 194 . . . . . . . . . . . 12  |-  ( ( A  i^i  C )  =/=  (/)  ->  -.  ( A  i^i  C )  =  (/) )
32 iffalse 3938 . . . . . . . . . . . 12  |-  ( -.  ( A  i^i  C
)  =  (/)  ->  if ( ( A  i^i  C )  =  (/) ,  (/) ,  B )  =  B )
3329, 31, 323syl 20 . . . . . . . . . . 11  |-  ( ( C  =/=  (/)  /\  C  C_  A )  ->  if ( ( A  i^i  C )  =  (/) ,  (/) ,  B )  =  B )
3422, 33syl5eq 2507 . . . . . . . . . 10  |-  ( ( C  =/=  (/)  /\  C  C_  A )  ->  (
( A  X.  B
) " C )  =  B )
3534difeq1d 3607 . . . . . . . . 9  |-  ( ( C  =/=  (/)  /\  C  C_  A )  ->  (
( ( A  X.  B ) " C
)  \  B )  =  ( B  \  B ) )
36 difid 3884 . . . . . . . . 9  |-  ( B 
\  B )  =  (/)
3735, 36syl6eq 2511 . . . . . . . 8  |-  ( ( C  =/=  (/)  /\  C  C_  A )  ->  (
( ( A  X.  B ) " C
)  \  B )  =  (/) )
3821, 37syl5sseq 3537 . . . . . . 7  |-  ( ( C  =/=  (/)  /\  C  C_  A )  ->  (
( ( R  i^i  ( A  X.  B
) ) " C
)  \  B )  C_  (/) )
39 ss0 3815 . . . . . . 7  |-  ( ( ( ( R  i^i  ( A  X.  B
) ) " C
)  \  B )  C_  (/)  ->  ( ( ( R  i^i  ( A  X.  B ) )
" C )  \  B )  =  (/) )
4038, 39syl 16 . . . . . 6  |-  ( ( C  =/=  (/)  /\  C  C_  A )  ->  (
( ( R  i^i  ( A  X.  B
) ) " C
)  \  B )  =  (/) )
41 df-ima 5001 . . . . . . . . . . 11  |-  ( ( R  \  ( A  X.  B ) )
" C )  =  ran  ( ( R 
\  ( A  X.  B ) )  |`  C )
42 df-res 5000 . . . . . . . . . . . 12  |-  ( ( R  \  ( A  X.  B ) )  |`  C )  =  ( ( R  \  ( A  X.  B ) )  i^i  ( C  X.  _V ) )
4342rneqi 5218 . . . . . . . . . . 11  |-  ran  (
( R  \  ( A  X.  B ) )  |`  C )  =  ran  ( ( R  \ 
( A  X.  B
) )  i^i  ( C  X.  _V ) )
4441, 43eqtri 2483 . . . . . . . . . 10  |-  ( ( R  \  ( A  X.  B ) )
" C )  =  ran  ( ( R 
\  ( A  X.  B ) )  i^i  ( C  X.  _V ) )
4544ineq1i 3682 . . . . . . . . 9  |-  ( ( ( R  \  ( A  X.  B ) )
" C )  i^i 
B )  =  ( ran  ( ( R 
\  ( A  X.  B ) )  i^i  ( C  X.  _V ) )  i^i  B
)
46 xpss1 5099 . . . . . . . . . . . 12  |-  ( C 
C_  A  ->  ( C  X.  _V )  C_  ( A  X.  _V )
)
47 sslin 3710 . . . . . . . . . . . 12  |-  ( ( C  X.  _V )  C_  ( A  X.  _V )  ->  ( ( R 
\  ( A  X.  B ) )  i^i  ( C  X.  _V ) )  C_  (
( R  \  ( A  X.  B ) )  i^i  ( A  X.  _V ) ) )
48 rnss 5220 . . . . . . . . . . . 12  |-  ( ( ( R  \  ( A  X.  B ) )  i^i  ( C  X.  _V ) )  C_  (
( R  \  ( A  X.  B ) )  i^i  ( A  X.  _V ) )  ->  ran  ( ( R  \ 
( A  X.  B
) )  i^i  ( C  X.  _V ) ) 
C_  ran  ( ( R  \  ( A  X.  B ) )  i^i  ( A  X.  _V ) ) )
4946, 47, 483syl 20 . . . . . . . . . . 11  |-  ( C 
C_  A  ->  ran  ( ( R  \ 
( A  X.  B
) )  i^i  ( C  X.  _V ) ) 
C_  ran  ( ( R  \  ( A  X.  B ) )  i^i  ( A  X.  _V ) ) )
5049adantl 464 . . . . . . . . . 10  |-  ( ( C  =/=  (/)  /\  C  C_  A )  ->  ran  ( ( R  \ 
( A  X.  B
) )  i^i  ( C  X.  _V ) ) 
C_  ran  ( ( R  \  ( A  X.  B ) )  i^i  ( A  X.  _V ) ) )
51 ssn0 3817 . . . . . . . . . . . 12  |-  ( ( C  C_  A  /\  C  =/=  (/) )  ->  A  =/=  (/) )
5251ancoms 451 . . . . . . . . . . 11  |-  ( ( C  =/=  (/)  /\  C  C_  A )  ->  A  =/=  (/) )
53 inss1 3704 . . . . . . . . . . . . . . . 16  |-  ( ( A  X.  _V )  i^i  R )  C_  ( A  X.  _V )
54 ssdif 3625 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  X.  _V )  i^i  R )  C_  ( A  X.  _V )  ->  ( ( ( A  X.  _V )  i^i 
R )  \  ( A  X.  B ) ) 
C_  ( ( A  X.  _V )  \ 
( A  X.  B
) ) )
5553, 54ax-mp 5 . . . . . . . . . . . . . . 15  |-  ( ( ( A  X.  _V )  i^i  R )  \ 
( A  X.  B
) )  C_  (
( A  X.  _V )  \  ( A  X.  B ) )
56 incom 3677 . . . . . . . . . . . . . . . 16  |-  ( ( A  X.  _V )  i^i  ( R  \  ( A  X.  B ) ) )  =  ( ( R  \  ( A  X.  B ) )  i^i  ( A  X.  _V ) )
57 indif2 3738 . . . . . . . . . . . . . . . 16  |-  ( ( A  X.  _V )  i^i  ( R  \  ( A  X.  B ) ) )  =  ( ( ( A  X.  _V )  i^i  R )  \ 
( A  X.  B
) )
5856, 57eqtr3i 2485 . . . . . . . . . . . . . . 15  |-  ( ( R  \  ( A  X.  B ) )  i^i  ( A  X.  _V ) )  =  ( ( ( A  X.  _V )  i^i  R ) 
\  ( A  X.  B ) )
59 difxp2 5418 . . . . . . . . . . . . . . 15  |-  ( A  X.  ( _V  \  B ) )  =  ( ( A  X.  _V )  \  ( A  X.  B ) )
6055, 58, 593sstr4i 3528 . . . . . . . . . . . . . 14  |-  ( ( R  \  ( A  X.  B ) )  i^i  ( A  X.  _V ) )  C_  ( A  X.  ( _V  \  B ) )
61 rnss 5220 . . . . . . . . . . . . . 14  |-  ( ( ( R  \  ( A  X.  B ) )  i^i  ( A  X.  _V ) )  C_  ( A  X.  ( _V  \  B ) )  ->  ran  ( ( R  \ 
( A  X.  B
) )  i^i  ( A  X.  _V ) ) 
C_  ran  ( A  X.  ( _V  \  B
) ) )
6260, 61mp1i 12 . . . . . . . . . . . . 13  |-  ( A  =/=  (/)  ->  ran  ( ( R  \  ( A  X.  B ) )  i^i  ( A  X.  _V ) )  C_  ran  ( A  X.  ( _V  \  B ) ) )
63 rnxp 5422 . . . . . . . . . . . . 13  |-  ( A  =/=  (/)  ->  ran  ( A  X.  ( _V  \  B ) )  =  ( _V  \  B
) )
6462, 63sseqtrd 3525 . . . . . . . . . . . 12  |-  ( A  =/=  (/)  ->  ran  ( ( R  \  ( A  X.  B ) )  i^i  ( A  X.  _V ) )  C_  ( _V  \  B ) )
65 disj2 3862 . . . . . . . . . . . 12  |-  ( ( ran  ( ( R 
\  ( A  X.  B ) )  i^i  ( A  X.  _V ) )  i^i  B
)  =  (/)  <->  ran  ( ( R  \  ( A  X.  B ) )  i^i  ( A  X.  _V ) )  C_  ( _V  \  B ) )
6664, 65sylibr 212 . . . . . . . . . . 11  |-  ( A  =/=  (/)  ->  ( ran  ( ( R  \ 
( A  X.  B
) )  i^i  ( A  X.  _V ) )  i^i  B )  =  (/) )
6752, 66syl 16 . . . . . . . . . 10  |-  ( ( C  =/=  (/)  /\  C  C_  A )  ->  ( ran  ( ( R  \ 
( A  X.  B
) )  i^i  ( A  X.  _V ) )  i^i  B )  =  (/) )
68 ssdisj 3864 . . . . . . . . . 10  |-  ( ( ran  ( ( R 
\  ( A  X.  B ) )  i^i  ( C  X.  _V ) )  C_  ran  ( ( R  \ 
( A  X.  B
) )  i^i  ( A  X.  _V ) )  /\  ( ran  (
( R  \  ( A  X.  B ) )  i^i  ( A  X.  _V ) )  i^i  B
)  =  (/) )  -> 
( ran  ( ( R  \  ( A  X.  B ) )  i^i  ( C  X.  _V ) )  i^i  B
)  =  (/) )
6950, 67, 68syl2anc 659 . . . . . . . . 9  |-  ( ( C  =/=  (/)  /\  C  C_  A )  ->  ( ran  ( ( R  \ 
( A  X.  B
) )  i^i  ( C  X.  _V ) )  i^i  B )  =  (/) )
7045, 69syl5eq 2507 . . . . . . . 8  |-  ( ( C  =/=  (/)  /\  C  C_  A )  ->  (
( ( R  \ 
( A  X.  B
) ) " C
)  i^i  B )  =  (/) )
71 disj3 3859 . . . . . . . 8  |-  ( ( ( ( R  \ 
( A  X.  B
) ) " C
)  i^i  B )  =  (/)  <->  ( ( R 
\  ( A  X.  B ) ) " C )  =  ( ( ( R  \ 
( A  X.  B
) ) " C
)  \  B )
)
7270, 71sylib 196 . . . . . . 7  |-  ( ( C  =/=  (/)  /\  C  C_  A )  ->  (
( R  \  ( A  X.  B ) )
" C )  =  ( ( ( R 
\  ( A  X.  B ) ) " C )  \  B
) )
7372eqcomd 2462 . . . . . 6  |-  ( ( C  =/=  (/)  /\  C  C_  A )  ->  (
( ( R  \ 
( A  X.  B
) ) " C
)  \  B )  =  ( ( R 
\  ( A  X.  B ) ) " C ) )
7440, 73uneq12d 3645 . . . . 5  |-  ( ( C  =/=  (/)  /\  C  C_  A )  ->  (
( ( ( R  i^i  ( A  X.  B ) ) " C )  \  B
)  u.  ( ( ( R  \  ( A  X.  B ) )
" C )  \  B ) )  =  ( (/)  u.  (
( R  \  ( A  X.  B ) )
" C ) ) )
7517, 74syl5eq 2507 . . . 4  |-  ( ( C  =/=  (/)  /\  C  C_  A )  ->  (
( R " C
)  \  B )  =  ( (/)  u.  (
( R  \  ( A  X.  B ) )
" C ) ) )
76 uncom 3634 . . . . 5  |-  ( (/)  u.  ( ( R  \ 
( A  X.  B
) ) " C
) )  =  ( ( ( R  \ 
( A  X.  B
) ) " C
)  u.  (/) )
77 un0 3809 . . . . 5  |-  ( ( ( R  \  ( A  X.  B ) )
" C )  u.  (/) )  =  (
( R  \  ( A  X.  B ) )
" C )
7876, 77eqtr2i 2484 . . . 4  |-  ( ( R  \  ( A  X.  B ) )
" C )  =  ( (/)  u.  (
( R  \  ( A  X.  B ) )
" C ) )
7975, 78syl6reqr 2514 . . 3  |-  ( ( C  =/=  (/)  /\  C  C_  A )  ->  (
( R  \  ( A  X.  B ) )
" C )  =  ( ( R " C )  \  B
) )
8079ancoms 451 . 2  |-  ( ( C  C_  A  /\  C  =/=  (/) )  ->  (
( R  \  ( A  X.  B ) )
" C )  =  ( ( R " C )  \  B
) )
8110, 80pm2.61dane 2772 1  |-  ( C 
C_  A  ->  (
( R  \  ( A  X.  B ) )
" C )  =  ( ( R " C )  \  B
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 367    = wceq 1398    =/= wne 2649   _Vcvv 3106    \ cdif 3458    u. cun 3459    i^i cin 3460    C_ wss 3461   (/)c0 3783   ifcif 3929    X. cxp 4986   ran crn 4989    |` cres 4990   "cima 4991
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-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
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-rab 2813  df-v 3108  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-br 4440  df-opab 4498  df-xp 4994  df-rel 4995  df-cnv 4996  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001
This theorem is referenced by: (None)
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